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]

Find the LCM for the given set of numbers.

$$6, 2, 22$$

Write an equation for the line with the given properties. Through (-1, 2) with slope $\frac{2}{3}$

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 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

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 $30 \mathrm{~s}$ ?

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

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

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

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

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

Determine the area of the shaded region in the following figures by integrating with respect to y.

$$x=y^2-3 y+12$$

$$x=-2 y^2-6 y+30$$

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 $x=-2 y^2-6 y+30$

Compute dy/dx for the following functions. y=

$$\tanh ^ { 2 }$$

x

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 $t=75$ ?