If necessary, use technology to evaluate or approximate the integral.

y=$\frac { 1 } { x }$ on [1, 10]

The velocity of a (fast) automobile on a straight highway is given by the function

$$v(t)= \begin{cases}3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45\end{cases}$$

where $t$ is measured in seconds and $v$ has units of $\mathrm{m} / \mathrm{s}$. What is the distance traveled by the automobile in the first $60 s$ ?

Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position. b. A spring that requires a force of 250 N to be stretched 0.5 from its equilibrium position.

Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.

$y=\frac{1}{\sqrt{1+x^2}}, y=0, x=-1$, and x=1

Write and simplify the integral that gives the arc length of the following curves on the given interval.

y=$\frac { 1 } { x }$ on [1, 10]

Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to find the volume of the following solids.
The region bounded by the curves $y=1+\sqrt{x}$ and $y=1-\sqrt{x}$, and the line $x=1$ is revolved about the $y$-axis. Find the volume of the resulting solid by (a) integrating with respect to $x$ and (b) integrating with respect to $y$. Be sure your answers agree.

Compound regions Sketch the following regions and then find the area.
The region bounded by $y=1-|x|$ and the $x$-axis

Sketch the following regions (if a figure is not given) and find the total area by integrating with respect to y.

The region bounded by $x=y^2-3 y+12$ and $x=-2 y^2-6 y+30$

The velocity of a (fast) automobile on a straight highway is given by the function

$$v(t)= \begin{cases}3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45\end{cases}$$

where $t$ is measured in seconds and $v$ has units of $\mathrm{m} / \mathrm{s}$. What is the distance traveled by the automobile in the first $30 \mathrm{~s}$ ?

Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. y=

$$9 x ^ { 2 / 3 } - \frac { x ^ { 4 / 3 } } { 32 }$$

, for

$$1 \leq x \leq 8$$

; about the x-axis

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the x-axis.

$y=\sqrt{\cos ^{-1} x}$, in the first quadrant

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.

$$y = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$

, y=0, x=-1, and x=1

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

$$v(t)= \begin{cases}3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45\end{cases}$$

where $t$ is measured in seconds and $v$ has units of $\mathrm{m} / \mathrm{s}$.
a. Graph the velocity function, for $0 \leq t \leq 70$. When is the velocity a maximum? When is the velocity zero? b. What is the distance traveled by the automobile in the first $30 \mathrm{~s}$ ? c. What is the distance traveled by the automobile in the first $60 \mathrm{~s}$ ? d. What is the position of the automobile when $t=75$ ?