Content check — Calculus Vol. 3, Chapter 5: Multiple Integration

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 5: Multiple Integration

246 problems · 86 figures

Checkpoint 5.1 §5.1 source ↗

Use the same function $z =$ $f (x, y) =$ $3 x^{2} -$ $y^{}$ over the rectangular region $R =$ $[0, 2] \times [0, 2] .$

Divide R into the same four squares with $m = n = 2,$ and choose the sample points as the upper left corner point of each square $(0, 1),$ $(1, 1),$ $(0, 2),$ and $(1, 2)$ (Figure 5.6) to approximate the signed volume of the solid S that lies above $R$ and “under” the graph of $f .$

On the xy plane, the points (1, 1), (1, 2), (2, 1), and (2, 2) are marked, and these form the upper right corners of four squares marked R11, R12, R21, and R22, respectively. Each square has area Delta A = 1. — Figure 5.6 in the book ↗

Answer

V =

\sum_{i = 1}^{2} \sum_{j = 1}^{2} f (x_{i j}^{*}, y_{i j}^{*}) Δ A =

0

Checkpoint 5.2 §5.1 source ↗

a. Use the properties of the double integral and Fubini’s theorem to evaluate the integral
$\int_{0}^{1} \int_{−1}^{3} (3 - x + 4 y) d y d x .$

b. Show that $0 \leq \iint_{R} sin π x cos π y d A \leq \frac{1}{32}$ where $R =$ $0, \frac{1}{4} \times \frac{1}{4}, \frac{1}{2} .$

Answer

a. 26 b. Answers may vary.

Checkpoint 5.3 §5.1 source ↗

Evaluate

\int_{y = −3}^{y = 2} \int_{x = 3}^{x = 5} (2 - 3 x^{2} + y^{2}) d x d y .

Answer

- \frac{1340}{3}

Checkpoint 5.4 §5.1 source ↗

Evaluate the integral

\iint_{R} x e^{x y} d A

where

R =

[0, 1] \times [0, ln 5] .

Answer

\frac{4 - ln 5}{ln 5}

Checkpoint 5.5 §5.1 source ↗

Find the volume of the solid bounded above by the graph of

f (x, y) =

x y sin (x^{2} y)

and below by the

x y

-plane on the rectangular region

R =

[0, 1] \times [0, π] .

Answer

\frac{π}{2}

Checkpoint 5.6 §5.1 source ↗

A contour map is shown for a function $f (x, y)$ on the rectangle $R =$ $[−3, 6] \times [−1, 4] .$

A contour map is shown with the highest point being about 18 and centered near (4, negative 1). From this point, the values decrease to 16, 14, 12, 10, 8, and 6 roughly every 0.5 to 1 distance. The lowest point is negative four near (negative 3, 4). There is a local minimum of 2 near (negative 1, 0).

a. Use the midpoint rule with $m = 3$ and $n = 2$ to estimate the value of $\iint_{R} f (x, y) d A .$

b. Estimate the average value of the function $f (x, y) .$

Answer

Answers to both parts a. and b. may vary.

5-1 §5.1 source ↗

In the following exercises, use the midpoint rule with $m = 4$ and $n = 2$ to estimate the volume of the solid bounded by the surface $z = f (x, y),$ the vertical planes $x = 1,$ $x = 2,$ $y = 1,$ and $y = 2,$ and the horizontal plane $z = 0 .$

$f (x, y) =$ $4 x +$ $2 y +$ $8 x y$

Answer

27.

5-3 §5.1 source ↗

In the following exercises, estimate $\iint_{R} f x, y d A$ by using a Riemann sum with $m = n = 2$ and the sample points to be the lower left corners of the subrectangles of the partition.

$f (x, y) =$ $sin x -$ $cos y,$ $R =$ $[0, π] \times [0, π]$

Answer

5-5 §5.1 source ↗

In the following exercises, estimate $\iint_{R} f x, y d A$ by using a Riemann sum with $m = n = 2$ and the sample points to be the lower left corners of the subrectangles of the partition.

Use the midpoint rule with $m = n = 2$ to estimate $\iint_{R} f (x, y) d A,$ where the values of the function f on $R =$ $[8, 10] \times [9, 11]$ are given in the following table.

	y
x	9	9.5	10	10.5	11
8	9.8	5	6.7	5	5.6
8.5	9.4	4.5	8	5.4	3.4
9	8.7	4.6	6	5.5	3.4
9.5	6.7	6	4.5	5.4	6.7
10	6.8	6.4	5.5	5.7	6.8

Answer

21.3.

5-7 §5.1 source ↗

In the following exercises, estimate $\iint_{R} f x, y d A$ by using a Riemann sum with $m = n = 2$ and the sample points to be the lower left corners of the subrectangles of the partition.

The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1-ft intervals, is given in the following table.

a. Estimate the volume of water in the swimming pool by using a Riemann sum with $m = n = 2 .$ Select the sample points using the midpoint rule on $R =$ $[0, 4] \times [0, 4] .$

b. Approximate the average depth of the swimming pool.

	y
x	0	1	2	3	4
0	1	1.5	2	2.5	3
1	1	1.5	2	2.5	3
2	1	1.5	1.5	2.5	3
3	1	1	1.5	2	2.5
4	1	1	1	1.5	2

Answer

a. 28

{ft}^{3}

b. 1.75 ft.

5-9 §5.1 source ↗

In the following exercises, estimate $\iint_{R} f x, y d A$ by using a Riemann sum with $m = n = 2$ and the sample points to be the lower left corners of the subrectangles of the partition.

The level curves $f (x, y) = k$ of the function f are given in the following graph, where k is a constant.

a. Apply the midpoint rule with $m = n = 2$ to estimate the double integral $\iint_{R} f (x, y) d A,$ where $R =$ $[0.2, 1] \times [0, 0.8] .$

b. Estimate the average value of the function f on R.
A series of curves marked k = negative 1, negative ½, negative ¼, negative 1/8, 0, 1/8, ¼, ½, and 1. The line marked k = 0 serves as an asymptote along the line y = x. The lines originate at (along the y axis) 1, 0.7, 0.5, 0.38, 0, (along the x axis) 0.38, 0.5, 0.7, and 1, with the further out lines curving less dramatically toward the asymptote.

Answer

0.112

f_{ave} ≃ 0.175;

here

f (0.4, 0.2) ≃ 0.1,

f (0.2, 0.6) ≃ −0.2,

f (0.8, 0.2) ≃ 0.6,

and

f (0.8, 0.6) ≃ 0.2 .

5-11 §5.1 source ↗

In the following exercises, estimate $\iint_{R} f x, y d A$ by using a Riemann sum with $m = n = 2$ and the sample points to be the lower left corners of the subrectangles of the partition.

The solid lying under the surface $z = \sqrt{4 - y^{2}}$ and above the rectangular region $R =$ $[0, 2] \times [0, 2]$ is illustrated in the following graph. Evaluate the double integral $\iint_{R} f (x, y) d A,$ where $f (x, y) =$ $\sqrt{4 - y^{2}},$ by finding the volume of the corresponding solid.

A quarter cylinder with center along the x axis and with radius 2. It has height 2 as shown.

Answer

2 π .

5-13 §5.1 source ↗

In the following exercises, calculate the integrals by interchanging the order of integration.

$\int_{−1}^{1} (\int_{−2}^{2} (2 x + 3 y + 5) d x) d y$

Answer

40.

5-15 §5.1 source ↗

In the following exercises, calculate the integrals by interchanging the order of integration.

$\int_{1}^{27} (\int_{1}^{2} (\sqrt[3]{x} + \sqrt[3]{y}) d y) d x$

Answer

\frac{81}{2} + 39 \sqrt[3]{2} .

5-17 §5.1 source ↗

In the following exercises, calculate the integrals by interchanging the order of integration.

$\int_{ln 2}^{ln 3} (\int_{0}^{l} e^{x + y} d y) d x$

Answer

e - 1 .

5-19 §5.1 source ↗

In the following exercises, calculate the integrals by interchanging the order of integration.

$\int_{1}^{6} (\int_{2}^{9} \frac{\sqrt{y}}{x^{2}} d y) d x$

Answer

15 - \frac{10 \sqrt{2}}{9} .

5-21 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{0}^{π} \int_{0}^{π / 2} sin (2 x) cos (3 y) d x d y$

Answer

5-23 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{1}^{e} \int_{1}^{e} [\frac{1}{x} sin (ln x) + \frac{1}{y} cos (ln y)] d x d y$

Answer

(e - 1) (1 + sin 1 - cos 1) .

5-25 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{1}^{2} \int_{1}^{2} (\frac{ln y}{x} + \frac{x}{2 y + 1}) d y d x$

Answer

\frac{3}{4} ln (\frac{5}{3}) +

2 {ln}^{2} 2 -

ln 2 .

5-27 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{1}^{\sqrt{3}} \int_{1}^{2} y arctan (\frac{1}{x}) d y d x$

Answer

\frac{1}{8} [(2 \sqrt{3} - 3) π + 6 ln 2] .

5-29 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{0}^{1} \int_{1}^{2} x e^{x + 4 y} d y d x$

Answer

\frac{1}{4} e^{4} (e^{4} - 1) .

5-31 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{1}^{e} \int_{1}^{e} (\frac{ln y}{\sqrt{y}} + \frac{ln x}{\sqrt{x}}) d y d x$

Answer

4 (e - 1) (2 - \sqrt{e}) .

5-33 §5.1 source ↗

In the following exercises, evaluate the iterated integrals by choosing the order of integration.

$\int_{0}^{1} \int_{1}^{2} (\frac{x}{x^{2} + y^{2}}) d y d x$

Answer

-

\frac{π}{4} +

ln (\frac{5}{4}) -

\frac{1}{2} ln 2 +

arctan 2 .

5-35 §5.1 source ↗

In the following exercises, find the average value of the function over the given rectangles.

$f (x, y) =$ $− x +$ $2 y,$ $R =$ $[0, 1] \times [0, 1]$

Answer

\frac{1}{2} .

5-37 §5.1 source ↗

In the following exercises, find the average value of the function over the given rectangles.

$f (x, y) =$ $sinh x +$ $sinh y,$ $R =$ $[0, 1] \times [0, 2]$

Answer

\frac{1}{2} 2 cosh 1 + cosh 2 - 3 .

5-49 §5.1 source ↗

In the following exercises, the function f is given in terms of double integrals.

[T] $f (x, y) =$ $\int_{0}^{y} \int_{0}^{x} (x s + y t) d s d t,$ where $(x, y) \in R =$ $[0, 1] \times [0, 1]$

Answer

a. $f (x, y) =$ $\frac{1}{2} x y (x^{2} + y^{2})$ b. $V =$ $\int_{0}^{1} \int_{0}^{1} f (x, y) d x d y =$ $\frac{1}{8}$ c. $f_{ave} = \frac{1}{8};$
d.

In xyz space, a plane is formed at z = 1/8, and there is another shape that starts at the origin, increases through the plane in a line roughly running from (1, 0.25, 0.125) to (0.25, 1, 0.125), and then rapidly increases to (1, 1, 1).

5-53 §5.1 source ↗

In the following exercises, the function f is given in terms of double integrals.

[T] Consider the function $f (x, y) =$ $e^{− x^{2} - y^{2}},$ where $(x, y) \in R =$ $[−1, 1] \times [−1, 1] .$

a. Use the midpoint rule with $m =$ $n =$ $2,$ $4,…, 10$ to estimate the double integral $I =$ $\iint_{R} e^{− x^{2} - y^{2}} d A .$ Round your answers to the nearest hundredths.

b. For $m = n = 2,$ find the average value of f over the region R. Round your answer to the nearest hundredths.

c. Use a CAS to graph in the same coordinate system the solid whose volume is given by $\iint_{R} e^{− x^{2} - y^{2}} d A$ and the plane $z = f_{ave} .$

Answer

a. For $m = n = 2,$ $I =$ $4 e^{−0.5} \approx$ $2.43$ b. $f_{ave} =$ $e^{−0.5} ≃ 0.61;$
c.

In xyz space, a plane is formed at z = 0.61, and there is another shape with maximum roughly at (0, 0, 0.92), which decreases along all the sides to the points (plus or minus 1, plus or minus 1, 0.12).

5-55 §5.1 source ↗

In the following exercises, the functions $f_{n}$ are given, where $n \geq 1$ is a natural number.

$f_{n} (x, y) =$ $x^{n} +$ $y^{n} +$ $x y,$ $(x, y) \in R =$ $[0, 1] \times [0, 1]$

Answer

\frac{2}{n + 1} + \frac{1}{4}

\frac{1}{4}

5-59 §5.1 source ↗

In the following exercises, the functions $f_{n}$ are given, where $n \geq 1$ is a natural number.

An isotherm map is a chart connecting points having the same temperature at a given time for a given period of time. Use the preceding exercise and apply the midpoint rule with $m = n = 2$ to find the average temperature over the region given in the following figure.

A contour map showing surface temperature in degrees Fahrenheit. Given the map, the midpoint rule would give rectangles with values 71, 72, 40, and 43.

Answer

56.5 °

F; here

f (x_{1}^{*}, y_{1}^{*}) =

71,

f (x_{2}^{*}, y_{1}^{*}) =

72,

f (x_{1}^{*}, y_{2}^{*}) =

40,

f (x_{2}^{*}, y_{2}^{*}) =

43,

where

x_{i}^{*}

and

y_{j}^{*}

are the midpoints of the subintervals of the partitions of

[a, b]

and

[c, d],

respectively.

Checkpoint 5.7 §5.2 source ↗

Consider the region in the first quadrant between the functions

y = 2 x

and

y = x^{2} .

Describe the region first as Type I and then as Type II.

Answer

Type I and Type II are expressed as

{(x, y) | 0 \leq x \leq 2, x^{2} \leq y \leq 2 x}

and

{(x, y) | 0 \leq y \leq 4, \frac{1}{2} y \leq x \leq \sqrt{y}},

respectively.

Checkpoint 5.8 §5.2 source ↗

Sketch the region

D

and evaluate the iterated integral

\iint_{D} x y d y d x

where

D

is the region bounded by the curves

y = cos x

and

y = sin x

in the interval

[−3 π / 4, π / 4] .

Answer

π / 4

Checkpoint 5.9 §5.2 source ↗

Consider the region bounded by the curves

y = ln x

and

y = e^{x}

in the interval

[1, 2] .

Decompose the region into smaller regions of Type II.

Answer

{(x, y) | 0 \leq y \leq \ln 2, 1 \leq x \leq e^{y}} \cup {(x, y) | \ln 2 \leq y \leq e, 1 \leq x \leq 2} \cup {(x, y) | e \leq y \leq e^{2}, ln y \leq x \leq 2}

Checkpoint 5.10 §5.2 source ↗

Redo Example 5.14 using a union of two Type II regions.

Answer

Same as in the example shown.

Checkpoint 5.11 §5.2 source ↗

Evaluate the iterated integral

\iint_{D} (x^{2} + y^{2}) d A

over the region

D

in the first quadrant between the functions

y = 2 x

and

y = x^{2} .

Evaluate the iterated integral by integrating first with respect to

y

and then integrating first with resect to

x .

Answer

\frac{216}{35}

Checkpoint 5.12 §5.2 source ↗

Find the volume of the solid bounded above by

f (x, y) =

10 -

2 x +

y

over the region enclosed by the curves

y = 0

and

y = e^{x},

where

x

is in the interval

[0, 1] .

Answer

\frac{e^{2}}{4} + 10 e - \frac{49}{4}

cubic units

Checkpoint 5.13 §5.2 source ↗

Find the area of a region bounded above by the curve

y = x^{3}

and below by

y = 0

over the interval

[0, 3] .

Answer

\frac{81}{4}

square units

Checkpoint 5.14 §5.2 source ↗

Find the average value of the function

f (x, y) = x y

over the triangle with vertices

(0, 0),

(1, 0) and (1, 3) .

Answer

\frac{3}{4}

Checkpoint 5.15 §5.2 source ↗

Evaluate the improper integral

\iint_{D} \frac{y}{\sqrt{1 - x^{2} - y^{2}}} d A

where

D =

{(x, y) | x \geq 0, y \geq 0, x^{2} + y^{2} \leq 1} .

Answer

\frac{π}{4}

Checkpoint 5.16 §5.2 source ↗

The joint density function for two random variables $X$ and $Y$ is given by

f (x, y) = \begin{matrix} \frac{1}{16250} (x^{2} + y^{2}) & if 0 \leq x \leq 15, 0 \leq y \leq 10 \\ 0 & otherwise \end{matrix}

Find the probability that $X$ is at most $10$ and $Y$ is at least $5 .$

Answer

\frac{11}{39} \approx 0.282

5-61 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

Find the average value of the function $f (x, y) = 3 x y$ on the region graphed in the previous exercise.

Answer

\frac{27}{20}

5-63 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

The region $D$ bounded by $y =$ $sin x,$ $y =$ $1 +$ $sin x,$ $x =$ $0,$ $and x =$ $\frac{π}{2}$ as given in the following figure.

A region is bounded by y = 1 + sin x, y = sin x, x = 0, and x = pi/2.

Answer

Type I but not Type II

5-65 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

Find the area of the region $D$ given in the previous exercise.

Answer

\frac{π}{2}

5-67 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

Find the volume of the solid under the graph of the function $f (x, y) = x y + 1$ and above the region in the figure in the previous exercise.

Answer

\frac{1}{6} (8 + 3 π)

5-69 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

Find the signed volume of the solid under the graph of the function $f (x, y) = x + y$ and above the region in the figure from the previous exercise.

Answer

\frac{1000}{3}

5-71 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

The region $D$ bounded by $y = 0$ and $y = x^{2} - 1$ as given in the following figure.

A region is bounded by y = 0 and y = negative 1 + x squared.

Answer

Type I and Type II

5-73 §5.2 source ↗

In the following exercises, specify whether the region is of Type I or Type II.

Let $D$ be the region bounded above by the curve of the equation $y = 4 - x^{2}$ and below by $y = cos x$ and the $x$ -axis. Explain why $D$ is neither of Type I nor II.

Answer

The region

D

is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions

g_{1} (x)

and

g_{2} (x) .

The region

D

is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions

h_{1} (y)

and

h_{2} (y) .

5-75 §5.2 source ↗

In the following exercises, evaluate the double integral $\iint_{D} f (x, y) d A$ over the region $D .$

$f (x, y) = 1$ and $D =$ ${(x, y) | 0 \leq x \leq \frac{π}{2}, sin x \leq y \leq 1 + sin x}$

Answer

\frac{π}{2}

5-77 §5.2 source ↗

In the following exercises, evaluate the double integral $\iint_{D} f (x, y) d A$ over the region $D .$

$f (x, y) = x y$ and $D =$ ${(x, y) | - 1 \leq y \leq 1, y^{2} - 1 \leq x \leq \sqrt{1 - y^{2}}}$

Answer

0

5-79 §5.2 source ↗

In the following exercises, evaluate the double integral $\iint_{D} f (x, y) d A$ over the region $D .$

$f (x, y) = − x + 1$ and $D$ is the triangular region with vertices $(0, 0),$ $(0, 2),$ $and (2, 2)$

Answer

\frac{2}{3}

5-81 §5.2 source ↗

Evaluate the iterated integrals.

$\int_{0}^{1} \int_{2 \sqrt{x}}^{2 \sqrt{x} + 1} (x y + 1) d y d x$

Answer

\frac{41}{20}

5-83 §5.2 source ↗

Evaluate the iterated integrals.

$\int_{1}^{2} \int_{− u^{2} - 1}^{− u} (8 u v) d v d u$

Answer

−63

5-85 §5.2 source ↗

Evaluate the iterated integrals.

$\int_{0}^{1 / 2} \int_{− \sqrt{1 - 4 y^{2}}}^{\sqrt{1 - 4 y^{2}}} 4 d x d y$

Answer

π

5-87 §5.2 source ↗

Evaluate the iterated integrals.

Let $D$ be the region bounded by $y = 1,$ $y = x,$ $y = ln x,$ and the $x$ -axis.

a. Show that $\iint_{D} y d A =$ $\int_{0}^{1} \int_{0}^{x} y d y d x +$ $\int_{1}^{e} \int_{ln x}^{1} y d y d x$ by dividing $D$ into two regions of Type I.

b. Evaluate the integral $\iint_{D} y d A .$

Answer

a. Answers may vary; b.

\frac{2}{3}

5-89 §5.2 source ↗

Evaluate the iterated integrals.

Let $D$ be the region bounded by $y = x^{2}, y = x + 2,$ and $y = − x .$

a. Show that $\iint_{D} x d A =$ $\int_{0}^{1} \int_{− y}^{\sqrt{y}} x d x d y +$ $\int_{1}^{4} \int_{y - 2}^{\sqrt{y}} x d x d y$ by dividing the region $D$ into two regions of Type II, where $D =$ ${(x, y) | y \geq x^{2}, y \geq - x, y \leq x + 2} .$

b. Evaluate the integral $\iint_{D} x d A .$

Answer

a. Answers may vary; b.

\frac{7}{3}

5-91 §5.2 source ↗

Evaluate the iterated integrals.

The region $D$ bounded by $y =$ $cos x,$ $y =$ $4 +$ $cos x,$ and $x = \pm \frac{π}{3}$ is shown in the following figure. Find the area $A (D)$ of the region $D .$

A region is bounded by y = cos x, y = 4 + cos x, x = negative 1, and x = 1.

Answer

\frac{8 π}{3}

5-93 §5.2 source ↗

Evaluate the iterated integrals.

Let $D$ be the region bounded by $y =$ $1,$ $y =$ $x,$ $y =$ $ln x,$ and the $x$ -axis. Find the area $A (D)$ of the region $D .$

Answer

e - \frac{3}{2}

5-95 §5.2 source ↗

Evaluate the iterated integrals.

Find the average value of the function $f (x, y) = − x + 1$ on the triangular region with vertices $(0, 0),$ $(0, 2),$ and $(2, 2) .$

Answer

\frac{1}{3}

5-97 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

$\int_{0}^{1} \int_{x - 1}^{1 - x} x d y d x$

Answer

\int_{0}^{1} \int_{x - 1}^{1 - x} x d y d x =

\int_{−1}^{0} \int_{0}^{y + 1} x d x d y +

\int_{0}^{1} \int_{0}^{1 - y} x d x d y =

\frac{1}{3}

5-99 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

$\int_{−1}^{1} \int_{− \sqrt{1 - y^{2}}}^{\sqrt{1 - y^{2}}} y d x d y$

Answer

\int_{−1}^{1} \int_{− \sqrt{1 - y^{2}}}^{\sqrt{1 - y^{2}}} y d x d y =

\int_{−1}^{1} \int_{− \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y d y d x =

0

5-101 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

The region $D$ is given in the following figure. Evaluate the double integral $\iint_{D} (x^{2} - y^{2}) d A$ by using the easier order of integration.

A region is bounded by y to the fourth power = 1 minus x and y to the fourth power = 1 + x.

Answer

\iint_{D} (x^{2} - y^{2}) d A =

\int_{−1}^{1} \int_{y^{4} - 1}^{1 - y^{4}} (x^{2} - y^{2}) d x d y =

\frac{464}{4095}

5-103 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Find the volume in the first octant of the solid under the plane $z = 3 x + y$ and above the region determined by $y = x^{7}$ and $y = x .$

Answer

\frac{4}{5}

5-105 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Find the volume of the solid under the surface $z = x^{3}$ and above the plane region bounded by $x =$ $sin y,$ $x =$ $− sin y,$ and $x = 1$ for values of $y$ between $y =$ $\frac{-π}{2} and y =$ $\frac{π}{2}$

Answer

\frac{5 π}{32}

5-109 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Find the volume of the solid situated in the first octant and bounded by the planes $x + 2 y = 1,$ $x =$ $0,$ $y =$ $0,$ $z =$ $4,$ $and z =$ $0 .$

Answer

1

5-111 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Find the volume of the solid bounded by the planes $x +$ $y =$ $1,$ $x -$ $y =$ $1,$ $x +$ $y =$ $−1,$ $x -$ $y =$ $−1,$ $z =$ $1 and z =$ $0 .$

Answer

2

5-113 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Let $S_{1} and S_{2}$ be the solids situated in the first octant under the planes $2 x + 2 y + z = 2$ and $x + y + z = 1,$ respectively, and let $S$ be the solid situated between $S_{1},$ $S_{2},$ $x =$ $0,$ $and y =$ $0 .$

a. Find the volume of the solid $S_{1} .$

b. Find the volume of the solid $S_{2} .$

c. Find the volume of the solid $S$ by subtracting the volumes of the solids $S_{1} and S_{2} .$

Answer

\frac{1}{3};

\frac{1}{6};

\frac{1}{6}

5-115 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Consider the plane $x +$ $y +$ $z =$ $2$ and the cylinder $x^{2} +$ $y^{2} =$ $4$ in the first octant.

a. Find the volume under the plane.

b. Find the volume inside the cylinder under the plane $z =$ $2$ .

c. Find the volume above the plane, inside the cylinder, and below the plane $z =$ $2$ .

Answer

\frac{4}{3};

2 π;

\frac{6 π - 4}{3}

5-117 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

[T] The region $D$ bounded by the curves $y =$ $cos x,$ $x =$ $0,$ $and y =$ $x^{3}$ is shown in the following figure. Use a graphing calculator or CAS to find the x-coordinates of the intersection points of the curves and to determine the area of the region $D .$ Round your answers to six decimal places.

A region is bounded by y = cos x, y = x cubed, and x = 0.

Answer

0 and 0.865474;

A (D) =

0.621135

5-119 §5.2 source ↗

In the following exercises, change the order of integration and evaluate the integral.

Consider $X and Y$ two random variables of probability densities $p_{1} (x)$ and $p_{2} (y),$ respectively. The random variables $X and Y$ are said to be independent if their joint density function is given by $p (x, y) =$ $p_{1} (x) p_{2} (y) .$ At a drive-thru restaurant, customers spend, on average, $3$ minutes placing their orders and an additional $5$ minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events $X$ and $Y .$ If the waiting times are modeled by the exponential probability densities

$\begin{matrix} p_{1} (x) = {\begin{matrix} \frac{1}{3} e^{− x / 3} & x \geq 0, \\ 0 & otherwise, \end{matrix} & and & p_{2} (y) = {\begin{matrix} \frac{1}{5} e^{− y / 5} & y \geq 0, \\ 0 & otherwise, \end{matrix} \end{matrix}$

respectively, the probability that a customer will spend less than 6 minutes in the drive-thru line is given by $P [X + Y \leq 6] =$ $\iint_{D} p (x, y) d x d y,$ where $D =$ ${(x, y)} | x \geq 0,$ $y \geq 0,$ $x +$ $y \leq 6} .$ Find $P [X + Y \leq 6]$ and interpret the result.

Answer

P [X + Y \leq 6] =

1 +

\frac{3}{2 e^{2}} -

\frac{5}{e^{6 / 5}} \approx

0.45;

there is a

45 %

chance that a customer will spend

6

minutes in the drive-thru line.

Checkpoint 5.17 §5.3 source ↗

Sketch the region

R =

{(r, θ) | 1 \leq r \leq 2, - \frac{π}{2} \leq θ \leq \frac{π}{2}},

and evaluate

\iint_{R} x d A .

Answer

\frac{14}{3}

Checkpoint 5.18 §5.3 source ↗

Evaluate the integral

\iint_{R} (4 - x^{2} - y^{2}) d A

where

R

is the circle of radius

2

on the

x y

-plane.

Answer

8 π

Checkpoint 5.19 §5.3 source ↗

Evaluate the integral

\iint_{D} r^{2} {sin}^{2} 2 θ r d r d θ where D = {(r, θ) | - \frac{π}{4} \leq θ \leq \frac{π}{4}, 0 \leq r \leq 2 \sqrt{cos 2 θ}} .

Answer

π / 4

Checkpoint 5.20 §5.3 source ↗

Use polar coordinates to find an iterated integral for finding the volume of the solid enclosed by the paraboloids

z = x^{2} + y^{2}

and

z = 16 - x^{2} - y^{2} .

Answer

V =

\int_{0}^{2 π} \int_{0}^{2 \sqrt{2}} (16 - 2 r^{2}) r d r d θ =

64 π

cubic units

Checkpoint 5.21 §5.3 source ↗

Find the area enclosed inside the cardioid

r = 3 - 3 sin θ

and outside the cardioid

r = 1 + sin θ .

Answer

A =

2 \int_{− π / 2}^{π / 6} \int_{1 + sin θ}^{3 - 3 sin θ} r d r d θ =

8 π +

9 \sqrt{3}

Checkpoint 5.22 §5.3 source ↗

Evaluate the integral

\iint_{ℝ^{2}} e^{−4 (x^{2} + y^{2})} d x d y .

Answer

\frac{π}{4}

5-123 §5.3 source ↗

In the following exercises, express the region $D$ in polar coordinates.

$D$ is the region between the circles of radius $4$ and radius $5$ centered at the origin that lies in the second quadrant.

Answer

D =

{(r, θ) | 4 \leq r \leq 5, \frac{π}{2} \leq θ \leq π}

5-125 §5.3 source ↗

In the following exercises, express the region $D$ in polar coordinates.

$D$ is the region bounded by the $x$ -axis and $y = \sqrt{2 - x^{2}} .$

Answer

D =

{(r, θ) | 0 \leq r \leq \sqrt{2}, 0 \leq θ \leq π}

5-127 §5.3 source ↗

In the following exercises, express the region $D$ in polar coordinates.

$D =$ ${(x, y) | x^{2} + y^{2} \leq 4 y}$

Answer

D =

{(r, θ) | 0 \leq r \leq 4 sin θ, 0 \leq θ \leq π}

5-129 §5.3 source ↗

In the following exercises, the graph of the polar rectangular region $D$ is given. Express $D$ in polar coordinates.

A sector of an annulus D is drawn between theta = pi/4 and theta = pi/2 with inner radius 3 and outer radius 5.

Answer

D =

{(r, θ) | 3 \leq r \leq 5, \frac{π}{4} \leq θ \leq \frac{π}{2}}

5-131 §5.3 source ↗

In the following exercises, the graph of the polar rectangular region $D$ is given. Express $D$ in polar coordinates.

A sector of an annulus D is drawn between theta = 3 pi/4 and theta = 5 pi/4 with inner radius 3 and outer radius 5.

Answer

D =

{(r, θ) | 3 \leq r \leq 5, \frac{3 π}{4} \leq θ \leq \frac{5 π}{4}}

5-133 §5.3 source ↗

In the following exercises, the graph of the polar rectangular region $D$ is given. Express $D$ in polar coordinates.

In the following graph, the region $D$ is bounded by $y = x$ and $y = x^{2} .$

A region D is drawn between y = x and y = x squared, which looks like a deformed lens, with the bulbous part below the straight part.

Answer

D =

{(r, θ) | 0 \leq r \leq tan θ sec θ, 0 \leq θ \leq \frac{π}{4}}

5-135 §5.3 source ↗

In the following exercises, evaluate the double integral $\iint_{R} f (x, y) d A$ over the polar rectangular region $D .$

$f (x, y) =$ $x +$ $y,$ $D =$ ${(r, θ) | 3 \leq r \leq 5, 0 \leq θ \leq 2 π}$

Answer

0

5-137 §5.3 source ↗

In the following exercises, evaluate the double integral $\iint_{R} f (x, y) d A$ over the polar rectangular region $D .$

$f (x, y) =$ $x^{4} +$ $y^{4},$ $D =$ ${(r, θ) | 1 \leq r \leq 2, \frac{3 π}{2} \leq θ \leq 2 π}$

Answer

\frac{63 π}{16}

5-139 §5.3 source ↗

In the following exercises, evaluate the double integral $\iint_{R} f (x, y) d A$ over the polar rectangular region $D .$

$f (x, y) =$ $x^{4} +$ $2 x^{2} y^{2} +$ $y^{4},$ where $D =$ ${(r, θ) | 3 \leq r \leq 4, \frac{π}{3} \leq θ \leq \frac{2 π}{3}} .$

Answer

\frac{3367 π}{18}

5-141 §5.3 source ↗

In the following exercises, evaluate the double integral $\iint_{R} f (x, y) d A$ over the polar rectangular region $D .$

$f (x, y) =$ $arctan (\frac{y}{x}),$ where $D =$ ${(r, θ) | 2 \leq r \leq 3, \frac{π}{4} \leq θ \leq \frac{π}{3}}$

Answer

\frac{35 π^{2}}{576}

5-143 §5.3 source ↗

In the following exercises, evaluate the double integral $\iint_{R} f (x, y) d A$ over the polar rectangular region $D .$

$\iint_{D} (e^{x^{2} + y^{2}} + x^{4} + 2 x^{2} y^{2} + y^{4}) arctan (\frac{y}{x}) d A, D =$ ${(r, θ) | 1 \leq r \leq 2, \frac{π}{4} \leq θ \leq \frac{π}{3}}$

Answer

\frac{7 π^{2}}{576} 21 - e^{2} + e^{4}

5-145 §5.3 source ↗

In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

$\int_{2}^{3} \int_{0}^{x} \frac{x}{\sqrt{x^{2} + y^{2}}} d y d x =$ $\int_{0}^{π / 4} \int_{2 sec θ}^{3 sec θ} r cos θ d r d θ$

Answer

\frac{5}{2} ln (1 + \sqrt{2})

5-147 §5.3 source ↗

$\int_{0}^{1} \int_{x^{2}}^{x} \frac{y}{\sqrt{x^{2} + y^{2}}} d y d x =$ $\int_{0}^{π / 4} \int_{0}^{tan θ sec θ} r sin θ d r d θ$

Answer

\frac{1}{6} (2 - \sqrt{2})

5-149 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{2} \int_{− \sqrt{4 - y^{2}}}^{\sqrt{4 - y^{2}}} {(x^{2} + y^{2})}^{2} d x d y$

Answer

\int_{0}^{π} \int_{0}^{2} r^{5} d r d θ =

\frac{32 π}{3}

5-151 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

$\int_{0}^{4} \int_{− \sqrt{16 - x^{2}}}^{\sqrt{16 - x^{2}}} sin (x^{2} + y^{2}) d y d x$

Answer

\int_{− π / 2}^{π / 2} \int_{0}^{4} r sin (r^{2}) d r d θ =

π {sin}^{2} 8

5-153 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

Find the area of the region $D$ bounded by the polar axis and the upper half of the cardioid $r = 1 + cos θ .$

Answer

\frac{3 π}{4}

5-155 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

Find the total area of the region enclosed by the four-leaved rose $r = sin 2 θ$ (see the figure in the previous exercise).

Answer

\frac{π}{2}

5-157 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

Find the area of the region $D,$ which is the region inside the disk $x^{2} + y^{2} \leq 4$ and to the right of the line $x = 1 .$

Answer

\frac{1}{3} (4 π - 3 \sqrt{3})

5-159 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

Determine the average value of the function $f (x, y) =$ $\sqrt{x^{2} + y^{2}}$ over the region $D$ bounded by the polar curve $r = 3 sin 2 θ,$ where $0 \leq θ \leq \frac{π}{2}$ (see the following graph).

The first-quadrant petal of the four-petal rose given by r = 3sin (2 theta) is shown.

Answer

\frac{16}{3 π}

5-161 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

Find the volume of the solid bounded by the paraboloid $z = 2 - 9 x^{2} - 9 y^{2}$ and the plane $z = 1 .$

Answer

\frac{π}{18}

5-163 §5.3 source ↗

In the following exercises, convert the integrals to polar coordinates and evaluate them.

a. Find the volume of the solid $S_{1}$ inside the unit sphere $x^{2} + y^{2} + z^{2} = 1$ and above the plane $z = 0 .$

b. Find the volume of the solid $S_{2}$ inside the double cone ${(z - 1)}^{2} =$ $x^{2} +$ $y^{2}$ and above the plane $z = 0 .$

c. Find the volume of the solid outside the double cone ${(z - 1)}^{2} =$ $x^{2} +$ $y^{2}$ and inside the sphere $x^{2} + y^{2} + z^{2} = 1 .$

Answer

\frac{2 π}{3};

\frac{π}{3};

\frac{π}{3}

5-165 §5.3 source ↗

For the following two exercises, consider a spherical ring, which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure).

A spherical ring is shown, that is, a sphere with a cylindrical hole going all the way through it.

A cylindrical hole of diameter $6$ cm is bored through a sphere of radius $5$ cm such that the axis of the cylinder passes through the center of the sphere. Find the volume of the resulting spherical ring.

Answer

\frac{256 π}{3} {cm}^{3}

5-167 §5.3 source ↗

Find the volume of the solid that lies under the paraboloid $z = x^{2} + y^{2},$ inside the cylinder $x^{2} + y^{2} = x,$ and above the plane $z = 0 .$

Answer

\frac{3 π}{32}

5-169 §5.3 source ↗

Find the volume of the solid that lies under the plane $2 x + y + 2 z = 8$ and above the unit disk $x^{2} + y^{2} = 1 .$

Answer

4 π

5-171 §5.3 source ↗

Use the information from the preceding exercise to calculate the integral $\iint_{D} {(x^{2} + y^{2})}^{3} d A,$ where $D$ is the unit disk.

Answer

\frac{π}{4}

5-173 §5.3 source ↗

Apply the preceding exercise to calculate the integral $\iint_{D} \frac{e^{\sqrt{x^{2} + y^{2}}}}{\sqrt{x^{2} + y^{2}}} d x d y,$ where $D$ is the annular region between the circles of radii $1$ and $2$ situated in the third quadrant.

Answer

\frac{1}{2} π e (e - 1)

5-175 §5.3 source ↗

Apply the preceding exercise to calculate the integral $\iint_{D} \frac{y^{2}}{x^{2}} d A,$ where $D =$ ${(r, θ) | 1 \leq r \leq 2, \frac{π}{6} \leq θ \leq \frac{π}{3}} .$

Answer

\sqrt{3} - \frac{π}{4}

5-177 §5.3 source ↗

Evaluate $\iint_{D} arctan (\frac{y}{x}) \sqrt{x^{2} + y^{2}} d A,$ where $D =$ ${(r, θ) | 2 \leq r \leq 3, \frac{π}{4} \leq θ \leq \frac{π}{3}} .$

Answer

\frac{133 π^{2}}{864}

Checkpoint 5.23 §5.4 source ↗

Evaluate the triple integral

\underset{B}{∭} z sin x cos y d V

where

B =

{(x, y, z) | 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, 1 \leq z \leq 3} .

Answer

\underset{B}{∭} z sin x cos y d V =

8

Checkpoint 5.24 §5.4 source ↗

Consider the solid sphere

E =

{(x, y, z) | x^{2} + y^{2} + z^{2} \leq 9} .

Write the triple integral

\underset{E}{∭} f (x, y, z) d V

for an arbitrary function

f

as an iterated integral. Then evaluate this triple integral with

f (x, y, z) = 1 .

Notice that this gives the volume of a sphere using a triple integral.

Answer

\underset{E}{∭} 1 d V =

\int_{x = −3}^{x = 3} \int_{y = − \sqrt{9 - x^{2}}}^{y = \sqrt{9 - x^{2}}} \int_{z = − \sqrt{9 - x^{2} - y^{2}}}^{z = \sqrt{9 - x^{2} - y^{2}}} 1 d z d y d x = 36 π .

Checkpoint 5.25 §5.4 source ↗

Write five different iterated integrals equal to the given integral

\int_{z = 0}^{z = 4} \int_{y = 0}^{y = 4 - z} \int_{x = 0}^{x = \sqrt{y}} f (x, y, z) d x d y d z .

Answer

(i)

\int_{z = 0}^{z = 4} \int_{x = 0}^{x = \sqrt{4 - z}} \int_{y = x^{2}}^{y = 4 - z} f (x, y, z) d y d x d z,

(ii)

\int_{y = 0}^{y = 4} \int_{z = 0}^{z = 4 - y} \int_{x = 0}^{x = \sqrt{y}} f (x, y, z) d x d z d y,

(iii)

\int_{y = 0}^{y = 4} \int_{x = 0}^{x = \sqrt{y}} \int_{z = 0}^{z = 4 - y} f (x, y, z) d z d x d y,

(iv)

\int_{x = 0}^{x = 2} \int_{y = x^{2}}^{y = 4} \int_{z = 0}^{z = 4 - y} f (x, y, z) d z d y d x,

(v)

\int_{x = 0}^{x = 2} \int_{z = 0}^{z = 4 - x^{2}} \int_{y = x^{2}}^{y = 4 - z} f (x, y, z) d y d z d x

Checkpoint 5.26 §5.4 source ↗

Find the average value of the function

f (x, y, z) =

x y z

over the cube with sides of length

4

units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.

Answer

f_{ave} = 8

5-181 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the rectangular solid box $B .$

$\underset{B}{∭} (2 x + 3 y^{2} + 4 z^{3}) d V,$ where $B =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq 2, 0 \leq z \leq 3}$

Answer

192

5-183 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the rectangular solid box $B .$

$\underset{B}{∭} (x cos y + z) d V,$ where $B =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq π, −1 \leq z \leq 1}$

Answer

0

5-185 §5.4 source ↗

In the following exercises, change the order of integration by integrating first with respect to $z,$ then $x,$ then $y .$

$\int_{0}^{1} \int_{1}^{2} \int_{2}^{3} (x^{2} + ln y + z) d x d y d z$

Answer

\int_{1}^{2} \int_{2}^{3} \int_{0}^{1} (x^{2} + ln y + z) d z d x d y =

\frac{35}{6} +

2 ln 2

5-187 §5.4 source ↗

In the following exercises, change the order of integration by integrating first with respect to $z,$ then $x,$ then $y .$

$\int_{−1}^{2} \int_{1}^{3} \int_{0}^{4} (x^{2} z + \frac{1}{y}) d x d y d z$

Answer

\int_{1}^{3} \int_{0}^{4} \int_{−1}^{2} (x^{2} z + \frac{1}{y}) d z d x d y =

64 +

12 ln 3

5-191 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region $E =$ ${(x, y, z) | a \leq x \leq b, h_{1} (x) \leq y \leq h_{2} (x), e \leq z \leq f} .$

$\underset{E}{∭} (2 x + 5 y + 7 z) d V,$ where $E =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq - x + 1, 1 \leq z \leq 2}$

Answer

\frac{77}{12}

5-193 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region $E =$ ${(x, y, z) | a \leq x \leq b, h_{1} (x) \leq y \leq h_{2} (x), e \leq z \leq f} .$

$\underset{E}{∭} (sin x + sin y) d V,$ where $E =$ ${(x, y, z) | 0 \leq x \leq \frac{π}{2}, − cos x \leq y \leq cos x, −1 \leq z \leq 1}$

Answer

2

5-195 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the indicated bounded region $E .$

$\underset{E}{∭} (x + 2 y z) d V,$ where $E =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq x, 0 \leq z \leq 5 - x - y}$

Answer

\frac{439}{120}

5-197 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the indicated bounded region $E .$

$\underset{E}{∭} y d V,$ where $E =$ ${(x, y, z) | - 1 \leq x \leq 1, − \sqrt{1 - x^{2}} \leq y \leq \sqrt{1 - x^{2}}, 0 \leq z \leq 1 - x^{2} - y^{2}}$

Answer

0

5-199 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region $E$ of the form $E =$ ${(x, y, z) | g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, e \leq z \leq f} .$

$\underset{E}{∭} x^{2} d V,$ where $E =$ ${(x, y, z) | 1 - y^{2} \leq x \leq y^{2} - 1, −1 \leq y \leq 1, 1 \leq z \leq 2}$

Answer

- \frac{64}{105}

5-201 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region $E$ of the form $E =$ ${(x, y, z) | g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, e \leq z \leq f} .$

$\underset{E}{∭} (x - y z) d V,$ where $E =$ ${(x, y, z) | - y^{6} \leq x \leq \sqrt{y}, 0 \leq y \leq 1, −1 \leq z \leq 1}$

Answer

\frac{11}{26}

5-203 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

E = {(x, y, z) | g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, u_{1} (x, y) \leq z \leq u_{2} (x, y)} .

$\underset{E}{∭} z d V,$ where $E =$ ${(x, y, z) | - y \leq x \leq y, 0 \leq y \leq 1, 0 \leq z \leq 1 - x^{4} - y^{4}}$

Answer

\frac{113}{450}

5-205 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

E = {(x, y, z) | g_{1} (y) \leq x \leq g_{2} (y), c \leq y \leq d, u_{1} (x, y) \leq z \leq u_{2} (x, y)} .

$\underset{E}{∭} (x - z) d V,$ where $E =$ ${(x, y, z) | - \sqrt{1 - y^{2}} \leq x \leq 0, 0 \leq y \leq \frac{1}{2}, 0 \leq z \leq 1 - x^{2} - y^{2}}$

Answer

\frac{- 609 - 216 \sqrt{3} - 80 π}{5760} \approx

- 0.21431

5-207 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

$E =$ ${(x, y, z) | (x, y) \in D, u_{1} (x, y) \leq z \leq u_{2} (x, y)},$ where $D$ is the projection of $E$ onto the $x y$ -plane.

$\iint_{D} (\int_{1}^{2} (x + z) d z) d A,$ where $D =$ ${(x, y) | x^{2} + y^{2} \leq 1}$

Answer

\frac{3 π}{2}

5-209 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

$E =$ ${(x, y, z) | (x, y) \in D, u_{1} (x, y) \leq z \leq u_{2} (x, y)},$ where $D$ is the projection of $E$ onto the $x y$ -plane.

$\iint_{D} (\int_{0}^{10 - x - y} (x + 2 z) d z) d A,$ where $D =$ ${(x, y) | y \geq 0, x \geq 0, x + y \leq 10}$

Answer

1250

5-211 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

$E =$ ${(x, y, z) | (x, y) \in D, u_{1} (x, y) \leq z \leq u_{2} (x, y)},$ where $D$ is the projection of $E$ onto the $x y$ -plane.

The solid $E$ bounded by $y^{2} +$ $z^{2} =$ $9,$ $z =$ $0,$ $x =$ $0,$ and $x = 5$ is shown in the following figure. Evaluate the integral $\underset{E}{∭} z d V$ by integrating first with respect to $z,$ then $y, and then x .$

A solid arching shape that reaches its maximum along the y axis with z = 3. The shape reach zero at y = plus or minus 3, and the graph is truncated at x = 0 and 5.

Answer

\int_{0}^{5} \int_{−3}^{3} \int_{0}^{\sqrt{9 - y^{2}}} z d z d y d x =

90

5-213 §5.4 source ↗

In the following exercises, evaluate the triple integrals over the bounded region

$E =$ ${(x, y, z) | (x, y) \in D, u_{1} (x, y) \leq z \leq u_{2} (x, y)},$ where $D$ is the projection of $E$ onto the $x y$ -plane.

[T] The volume of a solid $E$ is given by the integral $\int_{−2}^{0} \int_{x}^{0} \int_{0}^{x^{2} + y^{2}} d z d y d x .$ Use a computer algebra system (CAS) to graph $E$ and find its volume. Round your answer to two decimal places.

Answer

$V = 5.33$

A complex shape that starts at the origin and reaches its maximum at (negative 2, negative 2, 8). The shape is truncated by the x = y plane, the x = 0 plane, the y = negative 2 plane, the z = 0 plane, and a complex triangular-like shape with curved edges and sides (negative 2, negative 2, 8), (0, 0, 0), and (0, negative 2, 4).

5-215 §5.4 source ↗

In the following exercises, use two circular permutations of the variables $x, y, and z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x, y, and z$ is the arrangement of the numbers in one of the following orders: $y,$ $z,$ $and x or z,$ $x,$ $and y .$

$\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (x^{2} z^{2} + 1) d x d y d z$

Answer

\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (y^{2} z^{2} + 1) d z d x d y;

\int_{0}^{1} \int_{1}^{3} \int_{2}^{4} (x^{2} y^{2} + 1) d y d z d x

5-217 §5.4 source ↗

$\int_{0}^{1} \int_{− y}^{y} \int_{0}^{1 - x^{4} - y^{4}} e^{x} d z d x d y$

Answer

\int_{0}^{1} \int_{-z}^{z} \int_{0}^{1 - y^{4} - z^{4}} e^{x} d x d y d z;

\int_{0}^{1} \int_{-x}^{x} \int_{0}^{1 - z^{4} - x^{4}} \ln z d y d z d x

5-219 §5.4 source ↗

Set up the integral that gives the volume of the solid $E$ bounded by $y^{2} = x^{2} + z^{2}$ and $y = a,$ where $a > 0$ and $y \geq 0$

Answer

V =

\int_{- a}^{a} \int_{- \sqrt{a^{2} - z^{2}}}^{\sqrt{a^{2} - z^{2}}} \int_{\sqrt{x^{2} + z^{2}}}^{a^{2}} d y d x d z

5-221 §5.4 source ↗

Find the average value of the function $f (x, y, z) =$ $x +$ $y +$ $z$ over the parallelepiped determined by $x =$ $0,$ $x =$ $1,$ $y =$ $0,$ $y =$ $3,$ $z =$ $0,$ and $z = 5 .$

Answer

\frac{9}{2}

5-223 §5.4 source ↗

Find the volume of the solid $E$ that lies under the plane $x + y + z = 9$ and whose projection onto the $x y$ -plane is bounded by $x = \sqrt{y - 1}, x = 0,$ and $x + y = 7 .$

Answer

\frac{156}{5}

5-225 §5.4 source ↗

Consider the pyramid with the base in the $x y$ -plane of $[−2, 2] \times [−2, 2]$ and the vertex at the point $(0, 0, 8) .$

a. Show that the equations of the planes of the lateral faces of the pyramid are $4 y + z = 8,$ $4 y - z = −8,$ $4 x + z = 8,$ and $−4 x + z = 8 .$

b. Find the volume of the pyramid.

Answer

a. Answers may vary; b.

\frac{128}{3}

5-227 §5.4 source ↗

The solid $E$ bounded by the sphere of equation $x^{2} + y^{2} + z^{2} = r^{2}$ with $r > 0$ and located in the first octant is represented in the following figure.

The eighth of a sphere of radius 2 with center at the origin for positive x, y, and z.

a. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z,$ then with $y,$ and then with $x .$

b. Rewrite the integral in part a. as an equivalent integral in five other orders.

Answer

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d y d x;

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - y^{2}}} d z d x d y,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d x d z,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - x^{2}}} \int_{0}^{\sqrt{r^{2} - x^{2} - z^{2}}} d y d z d x,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - z^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d y d z,

\int_{0}^{r} \int_{0}^{\sqrt{r^{2} - y^{2}}} \int_{0}^{\sqrt{r^{2} - y^{2} - z^{2}}} d x d z d y

5-229 §5.4 source ↗

Find the volume of the prism with vertices $(0, 0, 0),$ $(2, 0, 0),$ $(2, 3, 0),$ $(0, 3, 0),$ $(0, 0, 1),$ $and (2, 0, 1) .$

Answer

3

5-231 §5.4 source ↗

The solid $E$ bounded by $z = 10 - 2 x - y$ and situated in the first octant is given in the following figure. Find the volume of the solid.

A tetrahedron bounded by the x y, y z, and x z planes and a triangle with vertices (0, 0, 10), (5, 0, 0), and (0, 10, 0).

Answer

\frac{250}{3}

5-233 §5.4 source ↗

The midpoint rule for the triple integral $\underset{B}{∭} f (x, y, z) d V$ over the rectangular solid box $B$ is a generalization of the midpoint rule for double integrals. The region $B$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} f (\bar{x_{i}}, \bar{y_{j}}, \bar{z_{k}}) Δ V,$ where $(\bar{x_{i}}, \bar{y_{j}}, \bar{z_{k}})$ is the center of the box $B_{i j k}$ and $Δ V$ is the volume of each subbox. Apply the midpoint rule to approximate $\underset{B}{∭} x^{2} d V$ over the solid $B =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

Answer

\frac{5}{16} \approx 0.313

5-235 §5.4 source ↗

Suppose that the temperature in degrees Celsius at a point $(x, y, z)$ of a solid $E$ bounded by the coordinate planes and $x + y + z = 5$ is $T (x, y, z) =$ $x z +$ $5 z +$ $10 .$ Find the average temperature over the solid.

Answer

\frac{35}{2}

Checkpoint 5.27 §5.5 source ↗

Evaluate the triple integral

\int_{θ = 0}^{θ = π} \int_{r = 0}^{r = 1} \int_{z = 0}^{z = 4} (r z sin θ) r d z d r d θ .

Answer

\frac{16}{3}

Checkpoint 5.28 §5.5 source ↗

Consider the region

E

inside the right circular cylinder with equation

r = 2 sin θ,

bounded below by the

r θ

-plane and bounded above by

z = 4 - y .

Set up a triple integral with a function

f (r, θ, z)

in cylindrical coordinates.

Answer

\underset{E}{∭} f (r, θ, z) r d z d r d θ =

\int_{θ = 0}^{θ = π} \int_{r = 0}^{r = 2 sin θ} \int_{z = 0}^{z = 4 - r sin θ} f (r, θ, z) r d z d r d θ .

Checkpoint 5.29 §5.5 source ↗

Redo the previous example with the order of integration

d θ d z d r .

Answer

E =

{(r, θ, z) | 0 \leq θ \leq 2 π, 0 \leq z \leq 1, z \leq r \leq 2 - z^{2}}

and

V =

\int_{r = 0}^{r = 1} \int_{z = r}^{z = 2 - r^{2}} \int_{θ = 0}^{θ = 2 π} r d θ d z d r .

Checkpoint 5.30 §5.5 source ↗

Redo the previous example with the order of integration

d θ d z d r .

Answer

E_{2} =

{(r, θ, z) | 0 \leq θ \leq 2 π, 0 \leq r \leq 1, r \leq z \leq \sqrt{4 - r^{2}}}

and

V =

\int_{r = 0}^{r = 1} \int_{z = r}^{z = \sqrt{4 - r^{2}}} \int_{θ = 0}^{θ = 2 π} r d θ d z d r .

Checkpoint 5.31 §5.5 source ↗

Set up a triple integral for the volume of the solid region bounded above by the sphere

ρ = 2

and bounded below by the cone

φ = π / 3 .

Answer

V (E) =

\int_{θ = 0}^{θ = 2 π} \int_{ϕ = 0}^{φ = π / 3} \int_{ρ = 0}^{ρ = 2} ρ^{2} sin φ d ρ d φ d θ

Checkpoint 5.32 §5.5 source ↗

Use rectangular, cylindrical, and spherical coordinates to set up triple integrals for finding the volume of the region inside the sphere $x^{2} + y^{2} + z^{2} = 4$ but outside the cylinder $x^{2} + y^{2} = 1 .$

A sphere with equation x squared + y squared + z squared = 4, and within it, a cylinder with equation x squared + y squared = 1.

Answer

Rectangular:

\int_{x = −2}^{x = 2} \int_{y = − \sqrt{4 - x^{2}}}^{y = \sqrt{4 - x^{2}}} \int_{z = − \sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}} d z d y d x - \int_{x = −1}^{x = 1} \int_{y = − \sqrt{1 - x^{2}}}^{y = \sqrt{1 - x^{2}}} \int_{z = − \sqrt{4 - x^{2} - y^{2}}}^{z = \sqrt{4 - x^{2} - y^{2}}} d z d y d x .

Cylindrical:

\int_{θ = 0}^{θ = 2 π} \int_{r = 1}^{r = 2} \int_{z = − \sqrt{4 - r^{2}}}^{z = \sqrt{4 - r^{2}}} r d z d r d θ .

Spherical:

\int_{φ = π / 6}^{φ = 5 π / 6} \int_{θ = 0}^{θ = 2 π} \int_{ρ = csc φ}^{ρ = 2} ρ^{2} sin φ d ρ d θ d φ .

5-241 §5.5 source ↗

In the following exercises, evaluate the triple integrals $\underset{B}{∭} f (x, y, z) d V$ over the solid $B .$

$f (x, y, z) = z,$ $B =$ ${(x, y, z) | x^{2} + y^{2} \leq 9, x \geq 0, y \geq 0, 0 \leq z \leq 1}$

A quarter section of a cylinder with height 1 and radius 3.

Answer

\frac{9 π}{8}

5-243 §5.5 source ↗

In the following exercises, evaluate the triple integrals $\underset{B}{∭} f (x, y, z) d V$ over the solid $B .$

$f (x, y, z) =$ $x y,$ $B =$ ${(x, y, z) | x^{2} + y^{2} \leq 1, x \geq 0, y \geq 0, x \leq y, −1 \leq z \leq 1}$

A wedge with radius 1, height 1, and angle pi/4.

Answer

\frac{1}{8}

5-245 §5.5 source ↗

In the following exercises, evaluate the triple integrals $\underset{B}{∭} f (x, y, z) d V$ over the solid $B .$

$f (x, y, z) =$ $e^{\sqrt{x^{2} + y^{2}}},$ $B =$ ${(x, y, z) | 1 \leq x^{2} + y^{2} \leq 4, y \leq 0, x \leq y \sqrt{3}, 2 \leq z \leq 3}$

Answer

\frac{π e^{2}}{6}

5-249 §5.5 source ↗

In the following exercises, the boundaries of the solid $E$ are given in cylindrical coordinates. Let $f (r, θ, z)$ be the corresponding function in cylindrical coordinates.

$E$ is inside the right circular cylinder $r = 4 sin θ,$ above the $r θ$ -plane, and inside the sphere $r^{2} + z^{2} = 16 .$

Answer

E =

{(r, θ, z) | 0 \leq θ \leq π, 0 \leq r \leq 4 sin θ, 0 \leq z \leq \sqrt{16 - r^{2}}};

\int_{0}^{π} \int_{0}^{4 sin θ} \int_{0}^{\sqrt{16 - r^{2}}} f (r, θ, z) r d z d r d θ

5-251 §5.5 source ↗

In the following exercises, the boundaries of the solid $E$ are given in cylindrical coordinates. Let $f (r, θ, z)$ be the corresponding function in cylindrical coordinates.

$E$ is located in the first octant and is bounded by the circular paraboloid $z = 9 - 3 r^{2},$ the cylinder $r = \sqrt{3},$ and the plane $r (cos θ + sin θ) =$ $20 -$ $z .$

Answer

E =

{(r, θ, z) | 0 \leq θ \leq \frac{π}{2}, 0 \leq r \leq \sqrt{3}, 9 - 3 r^{2} \leq z \leq 20 - r (cos θ + sin θ)};

\int_{0}^{π / 2} \int_{0}^{\sqrt{3}} \int_{9 - 3 r^{2}}^{20 - r (cos θ + sin θ)} f (r, θ, z) r d z d r d θ

5-253 §5.5 source ↗

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

$g (x, y, z) =$ $\frac{1}{x + 3},$ $E =$ ${(x, y, z) | 0 \leq x^{2} + y^{2} \leq 9, x \geq 0, y \geq 0, 0 \leq z \leq x + 3}$

Answer

E =

{(r, θ, z) | 0 \leq r \leq 3, 0 \leq θ \leq \frac{π}{2}, 0 \leq z \leq r cos θ + 3},

f (r, θ, z) =

\frac{1}{r cos θ + 3};

\int_{0}^{3} \int_{0}^{π / 2} \int_{0}^{r cos θ + 3} \frac{r}{r cos θ + 3} d z d θ d r = \frac{9 π}{4}

5-255 §5.5 source ↗

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

$g (x, y, z) = x,$ $E =$ ${(x, y, z) | 1 \leq y^{2} + z^{2} \leq 9, 0 \leq x \leq 9 - y^{2} - z^{2}}$

Answer

y =

r cos θ,

z =

r sin θ,

x =

z,

E =

{(r, θ, z) | 1 \leq r \leq 3, 0 \leq θ \leq 2 π, 0 \leq z \leq 9 - r^{2}},

f (r, θ, z) =

z;

\int_{1}^{3} \int_{0}^{2 π} \int_{0}^{9 - r^{2}} z r d z d θ d r = \frac{256 π}{3}

5-257 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E$ is above the $x y$ -plane, inside the cylinder $x^{2} + y^{2} = 1,$ and below the plane $z = 1 .$

Answer

π

5-259 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E$ is bounded by the circular cone $z = \sqrt{x^{2} + y^{2}}$ and $z = 1 .$

Answer

\frac{π}{3}

5-261 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E$ is located inside the cylinder $x^{2} + y^{2} = 1$ and between the circular paraboloids $z = 1 - x^{2} - y^{2}$ and $z = x^{2} + y^{2} .$

Answer

\frac{π}{4}

5-263 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E$ is located inside the circular cone $x^{2} +$ $y^{2} =$ ${(z - 1)}^{2}$ and between the planes $z = 0$ and $z = 2 .$

Answer

\frac{2 π}{3}

5-265 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

[T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $\int_{− π / 2}^{π / 2} \int_{0}^{1} \int_{r^{2}}^{r} r d z d r d θ .$ Find the volume $V$ of the solid. Round your answer to four decimal places.

Answer

$V =$ $\frac{π}{12} \approx$ $0.2618$

A quarter section of an ellipsoid with width 2, height 1, and depth 1.

5-267 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

Convert the integral $\int_{0}^{1} \int_{− \sqrt{1 - y^{2}}}^{\sqrt{1 - y^{2}}} \int_{x^{2} + y^{2}}^{\sqrt{x^{2} + y^{2}}} x z d z d x d y$ into an integral in cylindrical coordinates.

Answer

\int_{0}^{1} \int_{0}^{π} \int_{r^{2}}^{r} z r^{2} cos θ d z d θ d r

5-269 §5.5 source ↗

In the following exercises, evaluate the triple integral $\underset{B}{∭} f (x, y, z) d V$ over the solid $B .$

$f (x, y, z) = 1,$ $B =$ ${(x, y, z) | x^{2} + y^{2} + z^{2} \leq 90, z \geq 0}$

A filled-in half-sphere with radius 3 times the square root of 10.

Answer

180 π \sqrt{10}

5-271 §5.5 source ↗

In the following exercises, evaluate the triple integral $\underset{B}{∭} f (x, y, z) d V$ over the solid $B .$

$f (x, y, z) =$ $\sqrt{x^{2} + y^{2}},$ $B$ is bounded above by the half-sphere $x^{2} + y^{2} + z^{2} = 9$ with $z \geq 0$ and below by the cone $z^{2} = x^{2} + y^{2} .$

Answer

\frac{81 π (π - 2)}{16}

5-277 §5.5 source ↗

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

$g (x, y, z) =$ $x +$ $y;$ $E =$ ${(x, y, z) | 1 \leq x^{2} + y^{2} + z^{2} \leq 4, z \geq 0, y \geq 0}$

Answer

f (ρ, θ, φ) =

ρ sin φ (cos θ + sin θ),

E =

{(ρ, θ, φ) | 1 \leq ρ \leq 2, 0 \leq θ \leq π, 0 \leq φ \leq \frac{π}{2}};

\int_{0}^{π} \int_{0}^{π / 2} \int_{1}^{2} ρ^{3} {sin}^{2} φ cos θ + sin θ =

\frac{15 π}{8}

5-279 §5.5 source ↗

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

$g (x, y, z) = z;$ $E =$ ${(x, y, z) | x^{2} + y^{2} + z^{2} - 2 z \leq 0, \sqrt{x^{2} + y^{2}} \leq z}$

Answer

f (ρ, θ, φ) =

ρ cos φ;

E =

{(ρ, θ, φ) | 0 \leq ρ \leq 2 cos φ, 0 \leq θ \leq 2 π, 0 \leq φ \leq \frac{π}{4}};

\int_{0}^{2 π} \int_{0}^{π / 4} \int_{0}^{2 cos φ} ρ^{3} sin φ cos φ d ρ d φ d θ =

\frac{7 π}{6}

5-281 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E =$ ${(x, y, z) | x^{2} + y^{2} + z^{2} - 2 z \leq 0, \sqrt{x^{2} + y^{2}} \leq z}$

Answer

π

5-283 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

Use spherical coordinates to find the volume of the ball $ρ \leq 3$ that is situated between the cones $φ =$ $\frac{π}{4} and φ =$ $\frac{π}{3} .$

Answer

9 π (\sqrt{2} - 1)

5-285 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

Convert the integral $\int_{0}^{4} \int_{0}^{\sqrt{16 - x^{2}}} \int_{− \sqrt{16 - x^{2} - y^{2}}}^{\sqrt{16 - x^{2} - y^{2}}} {(x^{2} + y^{2} + z^{2})}^{2} d z d y d x$ into an integral in spherical coordinates.

Answer

\int_{0}^{π / 2} \int_{0}^{π} \int_{0}^{4} ρ^{6} sin φ d ρ d φ d θ

5-287 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates $\int_{π / 2}^{π} \int_{5 π / 6}^{π / 6} \int_{0}^{2} ρ^{2} sin φ d ρ d φ d θ .$ Find the volume $V$ of the solid. Round your answer to three decimal places.

Answer

$V =$ $\frac{4 π \sqrt{3}}{3} \approx$ $7.255$

A quarter section of an ellipsoid with width 2, height 2, and depth 1

5-289 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

[T] Use a CAS to evaluate the integral $\underset{E}{∭} (x^{2} + y^{2}) d V$ where $E$ lies above the paraboloid $z = x^{2} + y^{2}$ and below the plane $z = 3 y .$

Answer

\frac{243 π}{32}

5-291 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

Express the volume of the solid inside the sphere $x^{2} + y^{2} + z^{2} = 16$ and outside the cylinder $x^{2} + y^{2} = 4$ as triple integrals in cylindrical coordinates and spherical coordinates.

Answer

\int_{0}^{2 π} \int_{2}^{4} \int_{− \sqrt{16 - r^{2}}}^{\sqrt{16 - r^{2}}} r d z d r d θ;

\int_{π / 6}^{5 π / 6} \int_{0}^{2 π} \int_{2 csc φ}^{4} ρ^{2} sin φ d ρ d θ d φ

5-293 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by

$p (ρ, θ, φ) =$ $\frac{P_{0}}{ρ^{2}} {cos}^{2} θ {sin}^{4} φ,$ where $P_{0}$ is a constant with units in watts. The total power within a sphere $B$ of radius $r$ meters is defined as $P =$ $\underset{B}{∭} p (ρ, θ, φ) d V .$ Find the total power $P$ within a sphere of radius 20 meters.

Answer

P = \frac{64 P_{0} π}{3}

watts

5-295 §5.5 source ↗

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

A charge cloud contained in a sphere $B$ of radius r centimeters centered at the origin has its charge density given by $q (x, y, z) =$ $k \sqrt{x^{2} + y^{2} + z^{2}} \frac{μ C}{{cm}^{3}},$ where $k > 0 .$ The total charge contained in $B$ is given by $Q =$ $\underset{B}{∭} q (x, y, z) d V .$ Find the total charge $Q .$

Answer

Q = k r^{4} π μ C

Checkpoint 5.33 §5.6 source ↗

Consider the same region

R

as in the previous example, and use the density function

ρ (x, y) = \sqrt{x y} .

Find the total mass. Hint: Use trigonometric substitution

\sqrt{x} = \sqrt{3} \sin θ

and then use the power reducing formulas for trigonometric functions.

Answer

\frac{9 π}{8} kg

Checkpoint 5.34 §5.6 source ↗

Consider the same lamina

R

as above, and use the density function

ρ (x, y) = \sqrt{x y} .

Find the moments

M_{x}

and

M_{y} .

Answer

M_{x} = \frac{81 π}{64}

and

M_{y} = \frac{81 π}{64}

Checkpoint 5.35 §5.6 source ↗

Again use the same region

R

as above and the density function

ρ (x, y) = \sqrt{x y} .

Find the center of mass.

Answer

\overset{−}{x} =

\frac{M_{y}}{m} =

\frac{81 π / 64}{9 π / 8} =

\frac{9}{8}

and

\overset{−}{y} =

\frac{M_{x}}{m} =

\frac{81 π / 64}{9 π / 8} =

\frac{9}{8} .

Checkpoint 5.36 §5.6 source ↗

Calculate the mass, moments, and the center of mass of the region between the curves

y = x

and

y = x^{2}

with the density function

ρ (x, y) = x

in the interval

0 \leq x \leq 1 .

Answer

\overset{−}{x} =

\frac{M_{y}}{m} =

\frac{1 / 20}{1 / 12} =

\frac{3}{5}

and

\overset{−}{y} =

\frac{M_{x}}{m} =

\frac{1 / 24}{1 / 12} =

\frac{1}{2}

Checkpoint 5.37 §5.6 source ↗

Calculate the centroid of the region between the curves

y = x

and

y = \sqrt{x}

with uniform density in the interval

0 \leq x \leq 1 .

Answer

x_{c} =

\frac{M_{y}}{m} =

\frac{1 / 15}{1 / 6} =

\frac{2}{5} and y_{c} =

\frac{M_{x}}{m} =

\frac{1 / 12}{1 / 6} =

\frac{1}{2}

Checkpoint 5.38 §5.6 source ↗

Again use the same region

R

as above and the density function

ρ (x, y) = \sqrt{x y} .

Find the moments of inertia.

Answer

I_{x} =

\int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} y^{2} \sqrt{x y} d y d x =

\frac{64}{35}

and

I_{y} =

\int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} x^{2} \sqrt{x y} d y d x =

\frac{64}{35} .

Also,

I_{0} =

\int_{x = 0}^{x = 2} \int_{y = 0}^{y = x} (x^{2} + y^{2}) \sqrt{x y} d y d x =

\frac{128}{35} .

Checkpoint 5.39 §5.6 source ↗

Use the same region

R

from Example 5.61 and the density function

ρ (x, y) = \sqrt{x y} .

Find the radii of gyration with respect to the

x -axis,

the

y -axis,

and the origin.

Answer

R_{x} = \frac{6 \sqrt{35}}{35},

R_{y} = \frac{6 \sqrt{35}}{35},

and

R_{0} = \frac{6 \sqrt{70}}{35} .

Checkpoint 5.40 §5.6 source ↗

Consider the same region

Q

(Figure 5.70), and use the density function

ρ (x, y, z) =

x y^{2} z .

Find the mass.

In x y z space, the solid Q is shown with corners (0, 0, 0), (0, 0, 2), (0, 3, 0), and (6, 0, 0). Alternatively, you could consider the solid as being bounded by the x y, x z, and y z planes and the plane x + 2y + 3z = 6, forming an irregular tetrahedron. — Figure 5.70 in the book ↗

Answer

\frac{54}{35} = 1.543

Checkpoint 5.41 §5.6 source ↗

Consider the same region

Q

(Figure 5.70) and use the density function

ρ (x, y, z) =

x y^{2} z .

Find the center of mass.

Answer

(\frac{3}{2}, \frac{9}{8}, \frac{1}{2})

Checkpoint 5.42 §5.6 source ↗

Consider the same region

Q

(Figure 5.70), and use the density function

ρ (x, y, z) =

x y^{2} z .

Find the moments of inertia about the three coordinate planes.

Answer

The moments of inertia of the tetrahedron

Q

about the

y z -plane,

the

x z -plane,

and the

x y -plane

are

99 / 35,

36 / 7,

and 243 / 35,

respectively.

5-297 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the triangular region with vertices $(0, 0),$ $(0, 3),$ and $(6, 0);$ $ρ (x, y) =$ $x y .$

A right triangle bounded by the x and y axes and the line y = negative x/2 + 3.

Answer

\frac{27}{2}

5-299 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the rectangular region with vertices $(0, 0),$ $(0, 3),$ $(6, 3),$ and $(6, 0);$ $ρ (x, y) = \sqrt{x y} .$

A rectangle bounded by the x and y axes and the lines x = 6 and y = 3.

Answer

24 \sqrt{2}

5-301 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the trapezoidal region determined by the lines $y =$ $-$ $\frac{1}{4} x +$ $\frac{5}{2},$ $y =$ $0,$ $y =$ $2,$ and $x = 0;$ $ρ (x, y) = 3 x y .$

A trapezoid bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5.

Answer

76

5-303 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the disk of radius $2$ centered at $(1, 2);$ $ρ (x, y) =$ $x^{2} +$ $y^{2} -$ $2 x -$ $4 y +$ $5 .$

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0).

Answer

8 π

5-305 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the region enclosed by the ellipse $x^{2} +$ $4 y^{2} =$ $1;$ $ρ (x, y) =$ $1 .$

An ellipse with center the origin, major axis 2, and minor axis 0.5.

Answer

\frac{π}{2}

5-307 §5.6 source ↗

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $ρ .$

$R$ is the region bounded by $y =$ $x,$ $y =$ $− x,$ $y =$ $x +$ $2,$ $y =$ $− x +$ $2;$ $ρ (x, y) = 1 .$

A square with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1).

Answer

2

5-309 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions.

[T] $R$ is the triangular region with vertices $(0, 0),$ $(0, 3),$ and $(6, 0);$ $ρ (x, y) =$ $x y .$

Answer

a. $M_{x} =$ $\frac{81}{5},$ $M_{y} =$ $\frac{162}{5};$ b. $\overset{−}{x} =$ $\frac{12}{5},$ $\overset{−}{y} =$ $\frac{6}{5};$
c.

A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).

5-311 §5.6 source ↗

[T] $R$ is the rectangular region with vertices $(0, 0),$ $(0, 3),$ $(6, 3),$ $and (6, 0);$ $ρ (x, y) = \sqrt{x y} .$

Answer

a. $M_{x} =$ $\frac{216 \sqrt{2}}{5},$ $M_{y} =$ $\frac{432 \sqrt{2}}{5};$ b. $\overset{−}{x} =$ $\frac{18}{5},$ $\overset{−}{y} =$ $\frac{9}{5};$
c.

A rectangle R bounded by the x and y axes and the lines x = 6 and y = 3 with point marked (18/5, 9/5).

5-313 §5.6 source ↗

[T] $R$ is the trapezoidal region determined by the lines $y =$ $-$ $\frac{1}{4} x +$ $\frac{5}{2},$ $y =$ $0,$ $y = 2, and x = 0;$ $ρ (x, y) = 3 x y .$

Answer

a. $M_{x} =$ $\frac{368}{5},$ $M_{y} =$ $\frac{1552}{5};$ b. $\overset{−}{x} =$ $\frac{388}{95},$ $\overset{−}{y} =$ $\frac{92}{95};$
c.

A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).

5-315 §5.6 source ↗

[T] $R$ is the disk of radius $2$ centered at $(1, 2);$ $ρ (x, y) =$ $x^{2} +$ $y^{2} -$ $2 x -$ $4 y +$ $5 .$

Answer

a. $M_{x} =$ $16 π,$ $M_{y} =$ $8 π;$ b. $\overset{−}{x} = 1, \overset{−}{y} = 2;$
c.

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0) and has pointed marked at the center (1, 2).

5-317 §5.6 source ↗

[T] $R$ is the region enclosed by the ellipse $x^{2} +$ $4 y^{2} =$ $1;$ $ρ (x, y) =$ $1 .$

Answer

a. $M_{x} = 0, M_{y} = 0;$ b. $\overset{−}{x} = 0, \overset{−}{y} = 0;$
c.

An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.

5-319 §5.6 source ↗

[T] $R$ is the region bounded by $y =$ $x,$ $y =$ $− x,$ $y =$ $x +$ $2,$ and $y = − x + 2;$ $ρ (x, y) = 1 .$

Answer

a. $M_{x} = 2, M_{y} = 0;$ b. $\overset{−}{x} = 0, \overset{−}{y} = 1;$
c.

A square R with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1). A point is marked at (0, 1).

5-321 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the triangular region with vertices $(0, 0),$ $(0, 3),$ and $(6, 0);$ $ρ (x, y) =$ $x y .$

Answer

I_{x} =

\frac{243}{10},

I_{y} =

\frac{486}{5},

and I_{0} =

\frac{243}{2};

R_{x} =

\frac{3 \sqrt{5}}{5},

R_{y} =

\frac{6 \sqrt{5}}{5},

and R_{0} =

3

5-323 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the rectangular region with vertices $(0, 0),$ $(0, 3),$ $(6, 3),$ and $(6, 0);$ $ρ (x, y) =$ $\sqrt{x y} .$

Answer

I_{x} =

\frac{648 \sqrt{2}}{7},

I_{y} =

\frac{2592 \sqrt{2}}{7},

and I_{0} =

\frac{3240 \sqrt{2}}{7};

R_{y} =

\frac{3 \sqrt{21}}{7},

R_{x} =

\frac{6 \sqrt{21}}{7},

and R_{0} =

\frac{3 \sqrt{105}}{7}

5-325 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the trapezoidal region determined by the lines $y =$ $-$ $\frac{1}{4} x +$ $\frac{5}{2},$ $y =$ $0,$ $y =$ $2,$ and $x =$ $0;$ $ρ (x, y) =$ $3 x y .$

Answer

I_{x} =

88,

I_{y} =

1560,

and I_{0} =

1648;

R_{x} =

\frac{\sqrt{418}}{19},

R_{y} =

\frac{\sqrt{7410}}{19},

and

R_{0} = \frac{2 \sqrt{1957}}{19}

5-327 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the disk of radius $2$ centered at $(1, 2);$ $ρ (x, y) =$ $x^{2} +$ $y^{2} -$ $2 x -$ $4 y +$ $5 .$

Answer

I_{x} =

\frac{128 π}{3},

I_{y} =

\frac{56 π}{3},

and I_{0} =

\frac{184 π}{3};

R_{x} =

\frac{4 \sqrt{3}}{3},

R_{y} =

\frac{\sqrt{21}}{3},

and

R_{0} = \frac{\sqrt{69}}{3}

5-329 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the region enclosed by the ellipse $x^{2} +$ $4 y^{2} =$ $1;$ $ρ (x, y) =$ $1 .$

Answer

I_{x} =

\frac{π}{32},

I_{y} =

\frac{π}{8},

and I_{0} =

\frac{5 π}{32};

R_{x} =

\frac{1}{4},

R_{y} =

\frac{1}{2},

and R_{0} =

\frac{\sqrt{5}}{4}

5-331 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

$R$ is the region bounded by $y =$ $x,$ $y =$ $− x,$ $y =$ $x +$ $2,$ $and y =$ $− x +$ $2;$ $ρ (x, y) = 1 .$

Answer

I_{x} =

\frac{7}{3},

I_{y} =

\frac{1}{3},

and I_{0} =

\frac{8}{3};

R_{x} =

\frac{\sqrt{42}}{6},

R_{y} =

\frac{\sqrt{6}}{6},

and R_{0} =

\frac{2 \sqrt{3}}{3}

5-333 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

Let $Q$ be the solid unit cube. Find the mass of the solid if its density $ρ$ is equal to the square of the distance of an arbitrary point of $Q$ to the $x y -plane .$

Answer

m = \frac{1}{3}

5-337 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

[T] The solid $Q =$ ${(x, y, z) | x^{2} + y^{2} \leq 9, 0 \leq z \leq 1, x \geq 0, y \geq 0}$ has density equal to the distance to the $x y -plane .$ Use a CAS to answer the following questions.

a. Find the mass of $Q .$

b. Find the moments $M_{x y},$ $M_{x z},$ $and M_{y z}$ about the $x y -plane,$ $x z -plane,$ and $y z -plane,$ respectively.

c. Find the center of mass of $Q .$

d. Graph $Q$ and locate its center of mass.

Answer

a. $m = \frac{9 π}{8};$ b. $M_{x y} =$ $\frac{3 π}{4},$ $M_{x z} =$ $\frac{9}{2},$ $M_{y z} =$ $\frac{9}{2};$ c. $\overset{−}{x} =$ $\frac{4}{π},$ $\overset{−}{y} =$ $\frac{4}{π},$ $\overset{−}{z} =$ $\frac{2}{3};$ d. the solid $Q$ and its center of mass are shown in the following figure.

A quarter cylinder in the first quadrant with height 1 and radius 3. A point is marked at (9/(2 pi), 9/(2 pi), 2/3).

5-339 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

[T] The solid $Q$ has the mass given by the triple integral $\int_{−1}^{1} \int_{0}^{\frac{π}{4}} \int_{0}^{1} r^{2} d r d θ d z .$ Use a CAS to answer the following questions.

a. Show that the center of mass of $Q$ is located in the $x y -plane.$

b. Graph $Q$ and locate its center of mass.

Answer

a. $\overset{−}{x} =$ $\frac{3 \sqrt{2}}{2 π},$ $\overset{−}{y} =$ $\frac{3 (2 - \sqrt{2})}{2 π},$ $\overset{−}{z} =$ $0;$ b. the solid $Q$ and its center of mass are shown in the following figure.

A wedge from a cylinder in the first quadrant with height 2, radius 1, and angle roughly 45 degrees. A point is marked at (3 times the square root of 2/(2 pi), 3 times (2 minus the square root of 2)/(2 pi), 0).

5-343 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

The mass of a solid $Q$ is given by $\int_{0}^{2 \sqrt{2}} \int_{0}^{\sqrt{8 - x^{2}}} \int_{\sqrt{x^{2} + y^{2}}}^{\sqrt{16 - x^{2} - y^{2}}} {(x^{2} + y^{2} + z^{2})}^{n} d z d y d x,$ where $n$ is an integer. Determine $n$ such that the mass of the solid is $(2 - \sqrt{2}) π .$

Answer

n = −1

5-349 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

The solid $Q$ has the moment of inertia $I_{x}$ about the $y z -plane$ given by the triple integral $\int_{0}^{2} \int_{− \sqrt{4 - y^{2}}}^{\sqrt{4 - y^{2}}} \int_{\frac{1}{2} (x^{2} + y^{2})}^{\sqrt{x^{2} + y^{2}}} (y^{2} + z^{2}) (x^{2} + y^{2}) d z d x d y .$

a. Find the density of $Q .$

b. Find the moment of inertia $I_{z}$ about the $x y -plane.$

Answer

ρ (x, y, z) =

x^{2} +

y^{2};

\frac{16 π}{7}

5-351 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

Let $Q$ be the solid bounded by the $x y -plane,$ the cylinder $x^{2} + y^{2} = a^{2},$ and the plane $z = 1,$ where $a > 1$ is a real number. Find the moment $M_{x y}$ of the solid about the $x y -plane$ if its density given in cylindrical coordinates is $ρ (r, θ, z) =$ $\frac{d^{2} f}{d r^{2}} (r),$ where $f$ is a differentiable function with the first and second derivatives continuous and differentiable on $(0, a) .$

Answer

M_{x y} =

π (f (0) - f (a) + a f^{'} (a))

5-355 §5.6 source ↗

In the following exercises, consider a lamina occupying the region $R$ and having the density function $ρ$ given in the first two groups of Exercises.

Show that the moments of inertia $I_{x}, I_{y}, and I_{z}$ about the $y z -plane,$ $x z -plane,$ and $x y -plane,$ respectively, of the unit ball centered at the origin whose density is $ρ (x, y, z) =$ $e^{− x^{2} - y^{2} - z^{2}}$ are the same. Round your answer to two decimal places.

Answer

I_{x} =

I_{y} =

I_{z} ≃ 0.84

Checkpoint 5.43 §5.7 source ↗

Let a transformation

T

be defined as

T (u, v) =

(x, y)

where

x = u + v, y = 3 v .

Find the image of the rectangle

G =

{(u, v) : 0 \leq u \leq 1, 0 \leq v \leq 2}

from the

u v -plane

after the transformation into a region

R

in the

x y -plane .

Show that

T

is a one-to-one transformation and find

T^{−1} (x, y) .

Answer

T^{−1} (x, y) =

(u, v)

where

u = \frac{3 x - y}{3}

and

v = \frac{y}{3}

Checkpoint 5.44 §5.7 source ↗

Find the Jacobian of the transformation:

T (u, v) =

(u + v, 2 v) .

Answer

J (u, v) = \frac{\partial (x, y)}{\partial (u, v)} = | \begin{array}{l} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} | = | \begin{array}{l} 1 & 1 \\ 0 & 2 \end{array} | = 2

Checkpoint 5.45 §5.7 source ↗

Considering the integral

\int_{0}^{1} \int_{0}^{\sqrt{1 - x^{2}}} (x^{2} + y^{2}) d y d x,

use the change of variables

x = r cos θ

and

y = r sin θ,

and find the resulting integral.

Answer

\int_{0}^{π / 2} \int_{0}^{1} r^{3} d r d θ

Checkpoint 5.46 §5.7 source ↗

Make appropriate changes of variables in the integral

\iint_{R} \frac{4}{{(x - y)}^{2}} d y d x,

where

R

is the trapezoid bounded by the lines

x -

y =

2,

x -

y =

4,

x =

0,

and y =

0 .

Write the resulting integral.

Answer

x = \frac{1}{2} (v + u)

and

y = \frac{1}{2} (v - u)

and

\int_{2}^{4} \int_{−u}^{u} \frac{4}{u^{2}} (\frac{1}{2}) d v d u .

Checkpoint 5.47 §5.7 source ↗

Using the substitutions

x = v

and

y = \sqrt{u + v},

evaluate the integral

\iint_{R} y sin (y^{2} - x) d A

where

R

is the region bounded by the lines

y =

\sqrt{x},

x =

2,

and y =

0 .

Answer

\frac{1}{2} (sin 2 - 2)

Checkpoint 5.48 §5.7 source ↗

Let $D$ be the region in $x y z -space$ defined by $1 \leq x \leq 2,$ $0 \leq x y \leq 2,$ $and 0 \leq z \leq 1 .$

Evaluate $\underset{D}{∭} (x^{2} y + 3 x y z) d x d y d z$ by using the transformation $u = x, v = x y,$ and $w = 3 z .$

Answer

\int_{0}^{3} \int_{0}^{2} \int_{1}^{2} (\frac{v}{3} + \frac{v w}{3 u}) d u d v d w =

2 +

ln 8

5-357 §5.7 source ↗

In the following exercises, the function $T : S \to$ $R,$ $T (u, v) =$ $(x, y)$ on the region $S =$ ${(u, v) | 0 \leq u \leq 1, 0 \leq v \leq 1}$ bounded by the unit square is given, where $R \subset ℝ^{2}$ is the image of $S$ under $T .$

$x = \frac{u}{2}, y = \frac{v}{3}$

Answer

a. $T (u, v) =$ $(g (u, v), h (u, v)),$ $x =$ $g (u, v) =$ $\frac{u}{2}$ and $y =$ $h (u, v) =$ $\frac{v}{3} .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) = \frac{1}{2},$ $g_{v} (u, v) =$ $0,$ $h_{u} (u, v) =$ $0 and h_{v} (u, v) =$ $\frac{1}{3}$ are continuous on $S;$ b. $T (0, 0) =$ $(0, 0),$ $T (1, 0) =$ $(\frac{1}{2}, 0),$ $T (0, 1) =$ $(0, \frac{1}{3}),$ and $T (1, 1) =$ $(\frac{1}{2}, \frac{1}{3});$ c. $R$ is the rectangle of vertices $(0, 0),$ $(\frac{1}{2}, 0),$ $(\frac{1}{2}, \frac{1}{3}),$ $and (0, \frac{1}{3})$ in the $x y -plane;$ the following figure.

A rectangle with one corner at the origin, horizontal length 0.5, and vertical height 0.34.

5-359 §5.7 source ↗

$x =$ $2 u -$ $v,$ $y =$ $u +$ $2 v$

Answer

a. $T (u, v) =$ $(g (u, v), h (u, v)),$ $x =$ $g (u, v) =$ $2 u -$ $v,$ and $y =$ $h (u, v) =$ $u +$ $2 v .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) = 2,$ $g_{v} (u, v) = −1,$ $h_{u} (u, v) = 1,$ and $h_{v} (u, v) = 2$ are continuous on $S;$ b. $T (0, 0) =$ $(0, 0),$ $T (1, 0) =$ $(2, 1),$ $T (0, 1) =$ $(−1, 2),$ and $T (1, 1) =$ $(1, 3);$ c. $R$ is the sqaure of vertices $(0, 0),$ $(2, 1),$ $(1, 3),$ $and (−1, 2)$ in the $x y -plane;$ see the following figure.

A square of side length square root of 5 with one corner at the origin and another at (2, 1).

5-361 §5.7 source ↗

$x = u^{3}, y = v^{3}$

Answer

a. $T (u, v) =$ $(g (u, v), h (u, v)),$ $x =$ $g (u, v) =$ $u^{3},$ and $y =$ $h (u, v) =$ $v^{3} .$ The functions $g$ and $h$ are continuous and differentiable, and the partial derivatives $g_{u} (u, v) =$ $3 u^{2},$ $g_{v} (u, v) = 0,$ $h_{u} (u, v) = 0,$ and $h_{v} (u, v) = 3 v^{2}$ are continuous on $S;$ b. $T (0, 0) =$ $(0, 0),$ $T (1, 0) =$ $(1, 0),$ $T (0, 1) =$ $(0, 1),$ and $T (1, 1) =$ $(1, 1);$ c. $R$ is the unit square in the $x y -plane;$ see the following figure.

A Cartesian coordinate graph with axes ranging from 0 to 1 on both x (horizontal) and y (vertical) axes, displaying an empty plot area with no data or functions presented.

5-363 §5.7 source ↗

In the following exercises, determine whether the transformations $T : S \to R$ are one-to-one or not.

$x =$ $u^{4},$ $y =$ $u^{2} +$ $v,$ $where S$ is the triangle of vertices $(−2, 0),$ $(2, 0),$ $and (0, 2) .$

Answer

T

is not one-to-one: two points of

S

have the same image. Indeed,

T (−2, 0) =

T (2, 0) =

(16, 4) .

5-365 §5.7 source ↗

In the following exercises, determine whether the transformations $T : S \to R$ are one-to-one or not.

$T (u, v) =$ $(2 u - v, u),$ where $S$ is the triangle of vertices $(−1, 1),$ $(−1, −1),$ $and (1, −1) .$

Answer

T

is one-to-one: We argue by contradiction.

T (u_{1}, v_{1}) =

T (u_{2}, v_{2})

implies

2 u_{1} -

v_{1} =

2 u_{2} -

v_{2}

and

u_{1} = u_{2} .

Thus,

u_{1} = u_{2}

and

v_{1} = v_{2} .

5-367 §5.7 source ↗

In the following exercises, determine whether the transformations $T : S \to R$ are one-to-one or not.

$x =$ $u^{2} +$ $v +$ $w,$ $y =$ $u^{2} +$ $v,$ $z =$ $w,$ where $S = R = ℝ^{3} .$

Answer

T

is not one-to-one:

T (1, v, w) =

(−1, v, w)

5-369 §5.7 source ↗

In the following exercises, the transformations $T : S \to R$ are one-to-one. Find their related inverse transformations $T^{−1} : R \to S .$

$x =$ $u +$ $2 v,$ $y =$ $− u +$ $v,$ where $S = R = ℝ^{2} .$

Answer

u =

\frac{x - 2 y}{3},

v =

\frac{x + y}{3}

5-371 §5.7 source ↗

In the following exercises, the transformations $T : S \to R$ are one-to-one. Find their related inverse transformations $T^{−1} : R \to S .$

$x =$ $ln u,$ $y =$ $ln (u v),$ where $S =$ ${(u, v) | u > 0, v > 0}$ and $R = ℝ^{2} .$

Answer

u =

e^{x},

v =

e^{− x + y}

5-373 §5.7 source ↗

In the following exercises, the transformations $T : S \to R$ are one-to-one. Find their related inverse transformations $T^{−1} : R \to S .$

$x =$ $u +$ $v,$ $y =$ $v +$ $w,$ $z =$ $u +$ $w,$ where $S = R = ℝ^{3} .$

Answer

u =

\frac{x - y + z}{2},

v =

\frac{x + y - z}{2},

w =

\frac{− x + y + z}{2}

5-375 §5.7 source ↗

In the following exercises, the transformation $T : S \to$ $R,$ $T (u, v) =$ $(x, y)$ and the region $R \subset ℝ^{2}$ are given. Find the region $S \subset ℝ^{2} .$

$x =$ $a u,$ $y =$ $b v,$ $R =$ ${(x, y) | \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \leq 1},$ where $a, b > 0$

Answer

S =

{(u, v) | u^{2} + v^{2} \leq 1}

5-377 §5.7 source ↗

In the following exercises, the transformation $T : S \to$ $R,$ $T (u, v) =$ $(x, y)$ and the region $R \subset ℝ^{2}$ are given. Find the region $S \subset ℝ^{2} .$

$x =$ $a u,$ $y =$ $b v,$ $z =$ $c w,$ $R =$ ${(x, y) | \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} \leq 1, z > 0},$ where $a, b, c > 0$

Answer

R =

{(u, v, w) | u^{2} - v^{2} - w^{2} \leq 1, w > 0}

5-379 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

$x = \frac{u^{3}}{2}, y = \frac{v}{u^{2}}$

Answer

\frac{3}{2}

5-381 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

$x = u e^{v}, y = e^{− v}$

Answer

−1

5-383 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

$x =$ $v sin (u^{2}),$ $y =$ $v cos (u^{2})$

Answer

2 u v

5-385 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

$x =$ $v cosh (\frac{1}{u}),$ $y =$ $v sinh (\frac{1}{u}),$ $z =$ $u +$ $w^{2}$

Answer

\frac{2 v w}{u^{2}} or \frac{2 w u + w^{2}}{u^{2}}

5-387 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

$x =$ $u -$ $v,$ $y =$ $u +$ $v,$ $z =$ $u +$ $v +$ $w$

Answer

2

5-389 §5.7 source ↗

In the following exercises, find the Jacobian $J$ of the transformation.

The triangular region $R$ with the vertices $(0, 0),$ $(2, 0),$ $and (1, 3)$ is shown in the following figure.

A triangle with corners at the origin, (2, 0), and (1, 3).

a. Find a transformation $T : S \to R,$ $T (u, v) =$ $(x, y) =$ $(a u + b v, c u + d v),$ where $a, b, c$ and $d$ are real numbers with $a d - b c \neq 0$ such that $T^{−1} (0, 0) =$ $(0, 0),$ $T^{−1} (2, 0) =$ $(1, 0),$ and $T^{−1} (1, 3) =$ $(0, 1) .$

b. Use the transformation $T$ to find the area $A (R)$ of the region $R .$

Answer

T (u, v) =

(2 u + v, 3 v);

b. The area of

R

A (R) =

\int_{0}^{3} \int_{y / 3}^{(6 - y) / 3} d x d y = \int_{0}^{1} \int_{0}^{1 - u} | \frac{\partial (x, y)}{\partial (u, v)} | d v d u = \int_{0}^{1} \int_{0}^{1 - u} 6 d v d u = 3 .

5-391 §5.7 source ↗

In the following exercises, use the transformation $u = y - x, v = y,$ to evaluate the integrals on the parallelogram $R$ of vertices $(0, 0),$ $(1, 0),$ $(2, 1),$ $and (1, 1)$ shown in the following figure.

A rhombus with corners at the origin, (1, 0), (1, 1), and (2, 1).

$\iint_{R} (y^{2} - x y) d A$

Answer

- \frac{1}{4}

5-393 §5.7 source ↗

In the following exercises, use the transformation $y - x = u, x + y = v$ to evaluate the integrals on the square $R$ determined by the lines $y =$ $x,$ $y =$ $− x +$ $2,$ $y =$ $x +$ $2,$ and $y = − x$ shown in the following figure.

A square with side lengths square root of 2 rotated 45 degrees with one corner at the origin and another at (1, 1).

$\iint_{R} sin (x - y) d A$

Answer

−1 + cos 2

5-395 §5.7 source ↗

In the following exercises, use the transformation $x = u, 5 y = v$ to evaluate the integrals on the region $R$ bounded by the ellipse $x^{2} + 25 y^{2} = 1$ shown in the following figure.

An ellipse with center at the origin, major axis 2, and minor 0.4.

${\iint_{R} (x^{2} + 25 y^{2})}^{2} d A$

Answer

\frac{π}{15}

5-397 §5.7 source ↗

In the following exercises, use the transformation $u = x + y, v = x - y$ to evaluate the integrals on the trapezoidal region $R$ determined by the points $(1, 0),$ $(2, 0),$ $(0, 2),$ $and (0, 1)$ shown in the following figure.

A trapezoid with corners at (1, 0), (0, 1), (0, 2), and (2, 0).

$\iint_{R} (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) d A$

Answer

\frac{31}{5}

5-399 §5.7 source ↗

The solid $R$ bounded by the circular cylinder $x^{2} + y^{2} = 9$ and the planes $z = 0, z = 1,$ $x = 0, and y = 0$ is shown in the following figure. Find a transformation $T$ from a cylindrical box $S$ in $r θ z -space$ to the solid $R$ in $x y z -space .$

A quarter of a cylinder with height 1 and radius 3. The center axis is the z axis.

Answer

T (r, θ, z) =

(r cos θ, r sin θ, z);

S =

[0, 3] \times [0, \frac{π}{2}] \times [0, 1]

in the

r θ z -space

5-403 §5.7 source ↗

[T] Find the area of the region bounded by the curves $x y =$ $2,$ $x y =$ $3,$ $y =$ $x,$ and $y = 2 x$ by using the transformation $u = x y$ and $v = \frac{y}{x} .$

Answer

The area of the region is $\frac{\ln 2}{2} .$ A graph of the region is:

Four lines are drawn, namely, y = 3, y = 2, y = 3/(x squared), and y = 2/(x squared). The lines y = 3 and y = 2 are parallel to each other. The lines y = 3/(x squared) and y = 2/(x squared) are curves that run somewhat parallel to each other.

5-405 §5.7 source ↗

Evaluate the triple integral $\int_{0}^{2} \int_{4}^{6} \int_{3 z}^{3 z + 2} (5 - 4 y) d x d z d y$ by using the transformation $u =$ $x -$ $3 z,$ $v =$ $4 y,$ $and w =$ $z .$

Answer

8

5-409 §5.7 source ↗

[T] The transformations $T_{i} : ℝ^{2} \to ℝ^{2},$ $i = 1,…, 4,$ defined by $T_{1} (u, v) =$ $(u, − v),$ $T_{2} (u, v) =$ $(− u, v),$ $T_{3} (u, v) =$ $(− u, − v),$ and $T_{4} (u, v) =$ $(v, u)$ are called reflections about the $x -axis,$ $y -axis,$ origin, and the line $y = x,$ respectively.

a. Find the image of the region $S =$ ${(u, v) | u^{2} + v^{2} - 2 u - 4 v + 1 \leq 0}$ in the $x y -plane$ through the transformation $T_{1} \circ T_{2} \circ T_{3} \circ T_{4} .$

b. Use a CAS to graph the image of $S .$

c. Evaluate the integral $\iint_{S} sin (u^{2}) d u d v$ by using a CAS. Round your answer to two decimal places.

Answer

a. $R =$ ${(x, y) | y^{2} + x^{2} - 2 y - 4 x + 1 \leq 0};$ b. $R$ is graphed in the following figure;

A circle with radius 2 and center (2, 1).

c. $3.16$

5-411 §5.7 source ↗

[T] The transformation $T_{a, 0} : ℝ^{2} \to$ $ℝ^{2},$ $T_{a, 0} (u, v) =$ $(u + a v, v),$ where $a \neq 0$ is a real number, is called a shear in the $x -direction .$ The transformation, $T_{0, b} : ℝ^{2} \to$ $ℝ^{2},$ $T_{0, b} (u, v) =$ $(u, b u + v),$ where $b \neq 0$ is a real number, is called a shear in the $y -direction .$

a. Find transformations $T_{0, 2} \circ T_{3, 0} .$

b. Find the image $R$ of the trapezoidal region $S$ bounded by $u =$ $0,$ $v =$ $0,$ $v =$ $1,$ and $v = 2 - u$ through the transformation $T_{0, 2} \circ T_{3, 0} .$

c. Use a CAS to graph the image $R$ in the $x y -plane .$

d. Find the area of the region $R$ by using the area of region $S .$

Answer

a. $T_{0, 2} \circ T_{3, 0} (u, v) =$ $(u + 3 v, 2 u + 7 v);$ b. The image $R$ is the quadrilateral of vertices $(0, 0),$ $(3, 7),$ $(2, 4),$ $and (4, 9);$ c. $S$ is graphed in the following figure;

A four-sided figure with points the origin, (2, 4), (4, 9), and (3, 7).

d. $\frac{3}{2}$

5-413 §5.7 source ↗

Find the volume of a football whose shape is a spheroid $\frac{x^{2} + y^{2}}{a^{2}} +$ $\frac{z^{2}}{c^{2}} =$ $1$ whose length from tip to tip is $11$ inches and circumference at the center is $22$ inches. Round your answer to two decimal places.

Answer

\frac{2662}{3 π} ≃ 282.45 {in}^{3}

5-415 §5.7 source ↗

[T] Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation ${(\frac{x}{a})}^{n} +$ ${(\frac{y}{b})}^{n} =$ $1$ with $\frac{a}{b} = \frac{9}{7}$ and $n = e .$ Use a CAS to find an approximation of the area of the parking garage in the case $a = 900$ yards, $b = 700$ yards, and $n = 2.72$ .

Answer

A (R) =

2,

177,

216 {yd}^{2}

5-417 §5.R source ↗

True or False? Justify your answer with a proof or a counterexample.

Fubini’s theorem can be extended to three dimensions, as long as $f$ is continuous in all variables.

Answer

True.

5-419 §5.R source ↗

True or False? Justify your answer with a proof or a counterexample.

The Jacobian of the transformation $x =$ $u^{2} -$ $2 v,$ $y =$ $3 v -$ $2 u v$ is given by $−4 u^{2} + 6 u + 4 v .$

Answer

False.

5-421 §5.R source ↗

Evaluate the following integrals.

$\iint_{D} \frac{y}{3 x^{2} + 1} d A,$ $D =$ ${(x, y) | 0 \leq x \leq 1, − x \leq y \leq x}$

Answer

5-423 §5.R source ↗

Evaluate the following integrals.

$\int_{0}^{1} \int_{y}^{1} x y e^{x^{2}} d x d y$

Answer

\frac{1}{4}

5-425 §5.R source ↗

Evaluate the following integrals.

$\underset{R}{∭} 3 y d V,$ where $R =$ ${(x, y, z) | 0 \leq x \leq 1, 0 \leq y \leq x, 0 \leq z \leq \sqrt{9 - y^{2}}}$

Answer

1.475

5-427 §5.R source ↗

Evaluate the following integrals.

$\int_{0}^{2 π} \int_{0}^{π / 2} \int_{1}^{3} ρ^{2} sin (φ) d ρ d φ d θ$

Answer

\frac{52}{3} π

5-429 §5.R source ↗

For the following problems, find the specified area or volume.

The area of region enclosed by one petal of $r = cos (4 θ) .$

Answer

\frac{π}{16}

5-431 §5.R source ↗

For the following problems, find the specified area or volume.

The volume of the solid bounded by the cylinder $x^{2} + y^{2} = 16$ and from $z = 1$ and $z + x = 2 .$

Answer

16 π \approx 50.265

5-433 §5.R source ↗

For the following problems, find the center of mass of the region.

$ρ (x, y) = x y$ on the circle with radius $1$ in the first quadrant only.

Answer

(\frac{8}{15}, \frac{8}{15})

5-435 §5.R source ↗

For the following problems, find the center of mass of the region.

$ρ (x, y, z) = z$ on the inverted cone with radius $2$ and height $2 .$

Answer

(0, 0, \frac{4}{5})

5-437 §5.R source ↗

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height $1100$ ft and radius $6000$ ft.

If the compacted trash used to build Mount Holly on average has a density $400 {lb/ft}^{3},$ find the amount of work required to build the mountain.

Answer

1.452 π \times 10^{15}

ft-lb

5-439 §5.R source ↗

The following problems consider the temperature and density of Earth’s layers.

[T] The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree $3$ to the temperature along the radius of the Earth. Then find the average temperature of Earth. (Hint: begin at $0$ in the inner core and increase outward toward the surface)

Layer	Depth from center (km)	Temperature $° C$
Rocky Crust	0 to 40	0
Upper Mantle	40 to 150	870
Mantle	400 to 650	870
Inner Mantel	650 to 2700	870
Molten Outer Core	2890 to 5150	4300
Inner Core	5150 to 6378	7200

Source: http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Inside.shtml

Answer

y =

−1.238 \times 10^{−7} x^{3} +

0.001196 x^{2} -

3.666 x +

7208;

average temperature approximately

2800^{°} C

5-441 §5.R source ↗

The following problems concern the Theorem of Pappus (see Moments and Centers of Mass for a refresher), a method for calculating volume using centroids. Assuming a region $R,$ when you revolve around the $x -axis$ the volume is given by $V_{x} = 2 π A \bar{y},$ and when you revolve around the $y-axis$ the volume is given by $V_{y} = 2 π A \bar{x},$ where $A$ is the area of $R .$ Consider the region bounded by $x^{2} + y^{2} = 1$ and above $y = x + 1 .$

Find the volume when you revolve the region around the $x -axis.$

Answer

\frac{π}{3}

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Problems and answers are adapted from Calculus Volume 3 by Gilbert Strang and Edwin “Jed” Herman, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.