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

A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by $v(t)=9.8t$, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

How far does the probe fall in the first 30 s after it is released?

Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke's law is obeyed. A spring that requires $2 \mathrm{~J}$ of work to be stretched $0.1 \mathrm{~m}$ from its equilibrium position.

A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by $v(t)=9.8t$, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

Graph the velocity function.

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

y=4x-$x^2$ on [0, 4]

Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires a force of 50 N to be stretched 0.1 m from its equilibrium position. b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position.

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 $x=\cos y$ and $x=-\sin 2 y$.

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.

$$y = \frac { 1 } { \sqrt [ 4 ] { 1 - x ^ { 2 } } } , y = 0 , x = - \frac { 1 } { 2 }$$

, and x = $\frac { 1 } { 2 }$

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.

$y=\sqrt{\sin ^{-1} x}, y=\sqrt{\pi / 2}$, and x=0

Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions. What is the volume of the solid whose base is the region in the first quadrant bounded by y=$\sqrt { x }$, y=2-x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles?

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 position of the automobile when $t=75$ ?

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 $60 s$ ?

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

y=cos 2x on $[ 0 , \pi ]$

Calculate the work required to stretch the following springs 0.5 m from their equilibrium positions. Assume Hooke's law is obeyed. A spring that requires $50 \mathrm{~J}$ of work to be stretched $0.2 \mathrm{~m}$ from its equilibrium position.