Chapter 15: Multiple IntegrationContent check — Calculus (Guichard), Chapter 15: Multiple Integration

Compute ${\displaystyle {\int }_0^2{\int }_0^{4}1+xdydx}$⁠.

Compute ${\displaystyle {\int }_{-1}^1{\int }_0^{2}x+ydydx}$⁠.

Compute ${\displaystyle {\int }_1^2{\int }_0^{y}xydxdy}$⁠.

Compute ${\displaystyle {\int }_0^1{\int }_{y^2/2}^{\sqrt{y}}dxdy}$⁠.

Compute ${\displaystyle {\int }_1^2{\int }_1^x\frac{x^2}{y^2}dydx}$⁠.

Compute ${\displaystyle {\int }_0^1{\int }_0^{x^2}\frac{y}{e^x}dydx}$⁠.

Compute ${\displaystyle {\int }_0^{\sqrt{\pi /2}}{\int }_0^{x^2}x\cos ydydx}$⁠.

Compute ${\displaystyle {\int }_0^{\pi /2}}$ ${\displaystyle {\int }_0^{\cos \theta }{r}^2(\cos \theta -}$ ${\displaystyle r)drd\theta }$⁠.

Compute: ${\displaystyle {\int }_0^1{\int }_{\sqrt{y}}^1\sqrt{x^3+1}dxdy}$⁠.

Compute: ${\displaystyle {\int }_0^1{\int }_{y^2}^{1}y\sin (x^2)dxdy}$⁠.

Compute: ${\displaystyle {\int }_0^1{\int }_{x^2}^{1}x\sqrt{1+y^2}dydx}$

Compute: ${\displaystyle {\int }_0^1{\int }_0^y\frac{2}{\sqrt{1-x^2}}dxdy}$

Compute: ${\displaystyle {\int }_0^1{\int }_{3y}^3{e}^{x^2}dxdy}$

Compute ${\displaystyle {\int }_{-1}^1{\int }_0^{1-x^2}{x}^2-\sqrt{y}dydx}$⁠.

Compute ${\displaystyle {\int }_0^{\sqrt{2}/2}{\int }_{-\sqrt{1-2x^2}}^{\sqrt{1-2x^2}}xdydx}$⁠.

Evaluate over the region in the first quadrant bounded by the hyperbola $xy=16$ and the lines $y=x$⁠, $y=0$⁠, and $x=8$⁠.

Find the volume below $z=1-y$ above the region $-1\le x\le 1$⁠, $0\le y\le 1-x^2$⁠.

Find the volume bounded by $z=x^2+y^2$ and $z=4$⁠.

Find the volume in the first octant bounded by $y^2=4-x$ and $y=2z$⁠.

Find the volume in the first octant bounded by $y^2=4x$⁠, $2x+y=4$⁠, $z=y$⁠, and $y=0$⁠.

Find the volume in the first octant bounded by $x+y+z=9$⁠, $2x+3y=18$⁠, and $x+3y=9$⁠.

Find the volume in the first octant bounded by $x^2+y^2=a^2$ and $z=x+y$⁠.

Find the volume bounded by $4x^2+y^2=4z$ and $z=2$⁠.

Find the volume bounded by $z=x^2+y^2$ and $z=y$⁠.

Find the volume under the surface $z=xy$ above the triangle with vertices $(\mathrm{1,1,0})$⁠, $(\mathrm{4,1,0})$⁠, $(\mathrm{1,2,0})$⁠.

Find the volume enclosed by $y=x^2$⁠, $y=4$⁠, $z=x^2$⁠, $z=0$⁠.

A swimming pool is circular with a 40 meter diameter. The depth is constant along east-west lines and increases linearly from 2 meters at the south end to 7 meters at the north end. Find the volume of the pool.

Find the average value of $f(x,y)=e^y\sqrt{x+e^y}$ on the rectangle with vertices $(\mathrm{0,0})$⁠, $(\mathrm{4,0}$⁠), $(\mathrm{4,1})$ and $(\mathrm{0,1})$⁠.

Three cylinders of radius 1 intersect at right angles at the origin, as shown in figure 15.1.6. Find the volume contained inside all three cylinders.

Reverse the order of integration on each of the following integrals

a. ${\displaystyle {\int }_0^9{\int }_0^{\sqrt{9-y}}f(x,y)dxdy}$

b. ${\displaystyle {\int }_1^2{\int }_0^{\ln x}f(x,y)dydx}$

c. ${\displaystyle {\int }_0^1}$ ${\displaystyle {\int }_{\arcsin y}^{\pi /2}f(x,}$ ${\displaystyle y)dxdy}$

d. ${\displaystyle {\int }_0^1{\int }_{4x}^{4}f(x,y)dydx}$

e. ${\displaystyle {\int }_0^3{\int }_0^{\sqrt{9-y^2}}f(x,y)dxdy}$

Find the volume above the $x$⁠-⁠$y$ plane, under the surface $r^2=2z$⁠, and inside $r=2$⁠.

Find the volume inside both $r=1$ and $r^2+z^2=4$⁠.

Find the volume below $z=\sqrt{1-r^2}$ and above the top half of the cone $z=r$⁠.

Find the volume below $z=r$⁠, above the $x$⁠-⁠$y$ plane, and inside $r=\cos \theta$⁠.

Find the volume below $z=r$⁠, above the $x$⁠-⁠$y$ plane, and inside $r=1+\cos \theta$⁠.

Find the volume between $x^2+y^2=z^2$ and $x^2+y^2=z$⁠.

Find the area inside $r=1+\sin \theta$ and outside $r=2\sin \theta$⁠.

Find the area inside both $r=2\sin \theta$ and $r=2\cos \theta$⁠.

Find the area inside the four-leaf rose $r=\cos (2\theta )$ and outside $r=1/2$⁠.

Find the area inside the cardioid $r=2(1+\cos \theta )$ and outside $r=2$⁠.

Find the area of one loop of the three-leaf rose $r=\cos (3\theta )$⁠.

Compute ${\displaystyle {\int }_{-3}^3}$ ${\displaystyle {\int }_0^{\sqrt{9-x^2}}\sin (x^2+}$ ${\displaystyle y^2)dydx}$ by converting to cylindrical coordinates.

Compute ${\displaystyle {\int }_0^a{\int }_{-\sqrt{a^2-x^2}}^0{x}^{2}ydydx}$ by converting to cylindrical coordinates.

Find the volume under $z=y^2+x+2$ above the region $x^2+y^2\le 4$

Find the volume between $z=x^2{y}^3$ and $z=1$ above the region $x^2+y^2\le 1$

Find the volume inside $x^2+y^2=1$ and $x^2+z^2=1$⁠.

Find the volume under $z=r$ above $r=3+\cos \theta$⁠.

Consider the integral ⁠, where $D$ is the unit disk centered at the origin. (This is the same shape described in a different way in exercise 13 in section 9.7.)

a. Why might this integral be considered improper?

b. Calculate the value of the integral of the same function ${\displaystyle 1/\sqrt{x^2+y^2}}$ over the annulus with outer radius 1 and inner radius $\delta$⁠.

c. Obtain a value for the integral on the whole disk by letting $\delta$ approach 0.

Find the center of mass of a two-dimensional plate that occupies the square $[\mathrm{0,1}]\times [\mathrm{0,1}]$ and has density function $xy$⁠.

Find the center of mass of a two-dimensional plate that occupies the triangle $0\le x\le 1$⁠, $0\le y\le x$⁠, and has density function $xy$⁠.

Find the center of mass of a two-dimensional plate that occupies the upper unit semicircle centered at $(\mathrm{0,0})$ and has density function $y$⁠.

Find the center of mass of a two-dimensional plate that occupies the upper unit semicircle centered at $(\mathrm{0,0})$ and has density function $x^2$⁠.

Find the center of mass of a two-dimensional plate that occupies the triangle formed by $x=2$⁠, $y=x$⁠, and $y=2x$ and has density function $2x$⁠.

Find the center of mass of a two-dimensional plate that occupies the triangle formed by $x=0$⁠, $y=x$⁠, and $2x+y=6$ and has density function ${\displaystyle x^2}$⁠.

Find the center of mass of a two-dimensional plate that occupies the region enclosed by the parabolas $x=y^2$⁠, $y=x^2$ and has density function ${\displaystyle \sqrt{x}}$⁠.

Find the centroid of the area in the first quadrant bounded by $x^2-8y+4=0$⁠, $x^2=4y$⁠, and $x=0$⁠. (Recall that the centroid is the center of mass when the density is 1 everywhere.)

Find the centroid of one loop (say the loop along the $x$⁠-axis) of the three-leaf rose $r=\cos (3\theta )$⁠. (Recall that the centroid is the center of mass when the density is 1 everywhere, and that the mass in this case is the same as the area, which was the subject of exercise 11 in section 15.2.) The computations of the integrals for the moments $M_x$ and $M_y$ are elementary but quite long; Sage can help.

Find the center of mass of a two dimensional object that occupies the region $0\le x\le \pi$⁠, $0\le y\le \sin x$⁠, with density $\sigma =1$⁠.

A two-dimensional object has shape given by $r=1+\cos \theta$ and density $\sigma (r,\theta )=2+\cos \theta$⁠. Set up the three integrals required to compute the center of mass.

A two-dimensional object has shape given by $r=\cos \theta$ and density $\sigma (r,\theta )=r+1$⁠. Set up the three integrals required to compute the center of mass.

A two-dimensional object sits inside $r=1+\cos \theta$ and outside $r=\cos \theta$⁠, and has density $1$ everywhere. Set up the integrals required to compute the center of mass.

Find the area of the surface of a right circular cone of height $h$ and base radius $a$⁠.

Find the area of the portion of the plane $z=mx$ inside the cylinder $x^2+y^2=a^2$⁠.

Find the area of the portion of the plane $x+y+z=1$ in the first octant.

Find the area of the upper half of the cone $x^2+y^2=z^2$ inside the cylinder $x^2+y^2-2x=0$⁠.

Find the area of the upper half of the cone $x^2+y^2=z^2$ above the interior of one loop of $r=\cos (2\theta )$⁠.

Find the area of the upper hemisphere of $x^2+y^2+z^2=1$ above the interior of one loop of $r=\cos (2\theta )$⁠.

The plane $ax+by+cz=d$ cuts a triangle in the first octant provided that $a,b,c$ and $d$ are all positive. Find the area of this triangle.

Find the area of the portion of the cone $x^2+y^2=3z^2$ lying above the $xy$ plane and inside the cylinder $x^2+y^2=4y$⁠.

Evaluate ${\displaystyle {\int }_0^1{\int }_0^x{\int }_0^{x+y}2x+y-1dzdydx}$⁠.

Evaluate ${\displaystyle {\int }_0^2{\int }_{-1}^{x^2}{\int }_1^{y}xyzdzdydx}$⁠.

Evaluate ${\displaystyle {\int }_0^1}$ ${\displaystyle {\int }_0^x}$ ${\displaystyle {\int }_0^{\ln y}{e}^{x+y+z}dzdydx}$⁠.

Evaluate ${\displaystyle {\int }_0^{\pi /2}}$ ${\displaystyle {\int }_0^{\sin \theta }}$ ${\displaystyle {\int }_0^{r\cos \theta }{r}^{2}dzdrd\theta }$⁠.

Evaluate ${\displaystyle {\int }_0^{\pi }}$ ${\displaystyle {\int }_0^{\sin \theta }}$ ${\displaystyle {\int }_0^{r\sin \theta }r{\cos }^2\theta dzdrd\theta }$⁠.

Evaluate ${\displaystyle {\int }_0^1{\int }_0^{y^2}{\int }_0^{x+y}xdzdxdy}$⁠.

Evaluate ${\displaystyle {\int }_1^2}$ ${\displaystyle {\int }_y^{y^2}}$ ${\displaystyle {\int }_0^{\ln (y+z)}{e}^{x}dxdzdy}$⁠.

Compute ${\displaystyle {\int }_0^{\pi }{\int }_0^{\pi /2}{\int }_0^{1}z\sin x+}$ ${\displaystyle z\cos ydzdydx}$⁠.

Compute ${\displaystyle \int \int \int x+y+zdV}$ over the region $x^2+y^2+z^2\le 1$ in the first octant.

Find the mass of a cube with edge length 2 and density equal to the square of the distance from one corner.

Find the mass of a cube with edge length 2 and density equal to the square of the distance from one edge.

An object occupies the volume of the upper hemisphere of $x^2+y^2+z^2=4$ and has density $z$ at $(x,y,z)$⁠. Find the center of mass.

An object occupies the volume of the pyramid with corners at $(\mathrm{1,1,0})$⁠, $(1,-\mathrm{1,0})$⁠, $(-1,-\mathrm{1,0})$⁠, $(-\mathrm{1,1,0})$⁠, and $(\mathrm{0,0,2})$ and has density $x^2+y^2$ at $(x,y,z)$⁠. Find the center of mass.

Evaluate ${\displaystyle {\int }_0^1}$ ${\displaystyle {\int }_0^x}$ ${\displaystyle {\int }_0^{\sqrt{x^2+y^2}}\frac{{(x^2+y^2)}^{3/2}}{x^2+y^2+z^2}dzdydx}$⁠.

Evaluate ${\displaystyle {\int }_{-1}^1}$ ${\displaystyle {\int }_0^{\sqrt{1-x^2}}}$ ${\displaystyle {\int }_{\sqrt{x^2+y^2}}^{\sqrt{2-x^2-y^2}}\sqrt{x^2+y^2+z^2}dzdydx}$⁠.

Evaluate ${\displaystyle \int \int \int x^{2}dV}$ over the interior of the cylinder $x^2+y^2=1$ between $z=0$ and $z=5$⁠.

Evaluate ${\displaystyle \int \int \int xydV}$ over the interior of the cylinder $x^2+y^2=1$ between $z=0$ and $z=5$⁠.

Evaluate ${\displaystyle \int \int \int zdV}$ over the region above the $x$⁠-⁠$y$ plane, inside $x^2+y^2-2x=0$ and under $x^2+y^2+z^2=4$⁠.

Evaluate ${\displaystyle \int \int \int yzdV}$ over the region in the first octant, inside $x^2+y^2-2x=0$ and under $x^2+y^2+z^2=4$⁠.

Evaluate ${\displaystyle \int \int \int x^2+y^{2}dV}$ over the interior of $x^2+y^2+z^2=4$⁠.

Evaluate ${\displaystyle \int \int \int \sqrt{x^2+y^2}dV}$ over the interior of $x^2+y^2+z^2=4$⁠.

Compute ${\displaystyle \int \int \int x+y+zdV}$ over the region inside $x^2+y^2+z^2=1$ in the first octant.

Find the mass of a right circular cone of height $h$ and base radius $a$ if the density $k$ is proportional to the distance from the base.

Find the mass of a right circular cone of height $h$ and base radius $a$ if the density $k$ is proportional to the distance from its axis of symmetry.

An object occupies the region inside the unit sphere at the origin, and has density equal to the distance from the $x$⁠-axis. Find the mass.

An object occupies the region inside the unit sphere at the origin, and has density equal to the square of the distance from the origin. Find the mass.

An object occupies the region between the unit sphere at the origin and a sphere of radius 2 with center at the origin, and has density equal to the distance from the origin. Find the mass.

An object occupies the region in the first octant bounded by the cones $\varphi =\pi /4$ and $\varphi =\arctan 2$⁠, and the sphere $\rho =\sqrt{6}$⁠, and has density $k$ proportional to the distance from the origin. Find the mass.

Complete example 15.7.1 by converting to polar coordinates and evaluating the integral.

Evaluate over the square with corners $(\mathrm{0,0})$⁠, $(\mathrm{1,1})$⁠, $(\mathrm{2,0})$⁠, and $(1,-1)$ in two ways: directly, and using $x=(u+v)/2$⁠, $y=(u-v)/2$⁠.

Evaluate over the square with corners $(-\mathrm{1,0})$⁠, $(\mathrm{0,1})$⁠, $(\mathrm{1,0})$⁠, and $(0,-1)$ in two ways: directly, and using $x=(u+v)/2$⁠, $y=(u-v)/2$⁠.

Evaluate over the triangle with corners $(\mathrm{0,0})$⁠, $(-\mathrm{1,1})$⁠, and $(\mathrm{1,1})$ in two ways: directly, and using $x=(u+v)/2$⁠, $y=(u-v)/2$⁠.

Evaluate over the parallelogram with corners $(\mathrm{0,0})$⁠, $(\mathrm{3,3})$⁠, $(\mathrm{7,3})$⁠, and $(\mathrm{4,0})$ in two ways: directly, and using $x=u+v$⁠, $y=u$⁠.

Evaluate over the triangle with corners $(\mathrm{0,0})$⁠, $(\mathrm{4,4})$⁠, and $(\mathrm{4,0})$ using $x=u$⁠, $y=uv$⁠.

Evaluate over the region bounded by $xy=1$⁠, $xy=4$⁠, $y=1$⁠, and $y=4$ using $x=u/v$⁠, $y=v$⁠.

Evaluate over the region in the first quadrant bounded by the ellipse $9x^2+4y^2=1$⁠.

Compute the Jacobian for the substitutions $x=\rho \sin \varphi \cos \theta$⁠, $y=\rho \sin \varphi \sin \theta$⁠, $z=\rho \cos \varphi$⁠.

Evaluate where $E$ is the solid enclosed by the ellipsoid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,$$ using the transformation $x=au$⁠, $y=bv$⁠, and $z=cw$⁠.