Chapter 7: IntegrationContent check — Calculus (Guichard), Chapter 7: Integration

Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2t+2$⁠, and that at $t=1$ the object is at position 5. Find the position of the object at $t=2$⁠.

Suppose an object moves in a straight line so that its speed at time $t$ is given by ${\displaystyle v(t)=t^2+2}$⁠, and that at $t=0$ the object is at position 5. Find the position of the object at $t=2$⁠.

By a method similar to that in example 7.1.2, find the area under $y=2x$ between $x=0$ and any positive value for $x$⁠.

By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=0$ and any positive value for $x$⁠.

By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=2$ and any positive value for $x$ bigger than 2.

By a method similar to that in example 7.1.2, find the area under $y=4x$ between any two positive values for $x$⁠, say $a<b$⁠.

Let ${\displaystyle f(x)=x^2+3x+2}$⁠. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles.

Let ${\displaystyle f(x)=x^2-2x+3}$⁠. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles.

Find the antiderivatives of the functions:

${\displaystyle 8\sqrt{x}}$

Find the antiderivatives of the functions:

${\displaystyle 3t^2+1}$

Find the antiderivatives of the functions:

${\displaystyle 4/\sqrt{x}}$

Find the antiderivatives of the functions:

${\displaystyle 2/z^2}$

Find the antiderivatives of the functions:

${\displaystyle 7s^{-1}}$

Find the antiderivatives of the functions:

${\displaystyle {(5x+1)}^2}$

Find the antiderivatives of the functions:

${\displaystyle {(x-6)}^2}$

Find the antiderivatives of the functions:

${\displaystyle x^{3/2}}$

Find the antiderivatives of the functions:

${\displaystyle \frac{2}{x\sqrt{x}}}$

Find the antiderivatives of the functions:

${\displaystyle |2t-4|}$

Compute the values of the integrals:

${\displaystyle {\int }_1^4{t}^2+3tdt}$

Compute the values of the integrals:

${\displaystyle {\int }_0^{\pi }\sin tdt}$

Compute the values of the integrals:

${\displaystyle {\int }_1^{10}\frac{1}{x}dx}$

Compute the values of the integrals:

${\displaystyle {\int }_0^5{e}^{x}dx}$

Compute the values of the integrals:

${\displaystyle {\int }_0^3{x}^{3}dx}$

Compute the values of the integrals:

${\displaystyle {\int }_1^2{x}^{5}dx}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^x{t}^2-3tdt}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^{x^2}{t}^2-3tdt}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^x{e}^{t^2}dt}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^{x^2}{e}^{t^2}dt}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^x\tan (t^2)dt}$

Compute the values of the integrals:

Find the derivative of ${\displaystyle G(x)={\int }_1^{x^2}\tan (t^2)dt}$

An object moves so that its velocity at time $t$ is $v(t)=-9.8t+20$ m/s. Describe the motion of the object between $t=0$ and $t=5$⁠, find the total distance traveled by the object during that time, and find the net distance traveled.

An object moves so that its velocity at time $t$ is $v(t)=\sin t$⁠. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2\pi$⁠.

An object moves so that its velocity at time $t$ is $v(t)=1+2\sin t$ m/s. Find the net distance traveled by the object between $t=0$ and $t=2\pi$⁠, and find the total distance traveled during the same period.

Consider the function $f(x)=$ $(x+2)(x+1)(x-1)(x-2)$ on $[-\mathrm{2,2}]$⁠. Find the total area between the curve and the $x$⁠-axis (measuring all area as positive).

Consider the function ${\displaystyle f(x)=x^2-3x+2}$ on $[\mathrm{0,4}]$⁠. Find the total area between the curve and the $x$⁠-axis (measuring all area as positive).

Evaluate the three integrals: $$A={\int }_0^3(-x^2+9)dxB={\int }_0^4(-x^2+9)dxC={\int }_4^3(-x^2+9)dx,$$ and verify that $A=B+C$⁠.