Exam 6: Inverse Functions

In a certain city the temperature $7\left(\mathrm{I}_{0} \mathrm{ul}\right)$ hours after 7 A.M. was modeled by the function $T(t)=45+30 \sin \frac{\pi t}{12}$ Find the average temperature to three decimal places during the period from 7 A.M. to 7 P.M.

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = 2 $x^{2}$ , y = 4x - 2, y = 8; the y-axis

Find the number b such that the line $y=b$ divides the region bounded by the curves $y=x^{2}$ and $y=5$ into two regions with equal area.

The linear density of a $48$ m long rod is $\frac{72}{\sqrt{x+1}} \mathrm{~kg} / \mathrm{m}$ where x is measured in meters from one end of the rod. Find the average density of the rod.

Sketch a plane region, and indicate the axis about which it is revolved so that the resulting solid of revolution has the volume given by the integral. $\pi \int_{0}^{5}\left(25 x^{2}-x^{4}\right) d x$

Find the number(s) a such that the average value of the function $f(t)=25-20 t+3 t^{2}$ on the interval $[0, a]$ is equal to 4 .

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y=x^{2}, y=0, x=1, x=4 ; \text { about } x=1$

A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 11 cm to 24 cm?

Use cylindrical shells to find the volume of the solid. A sphere of radius $r$ .

Find the volume of the solid that is obtained by revolving the region about the x-axis.

The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = $\sqrt{16-x^{2}}$ , y = -x + 4; the y-axis

The velocity v of blood that flows in a blood vessel with radius $B$ and length l at a distance $b$ from the central axis is $v(r)=\frac{P}{4 q l}\left(B^{2}-b^{2}\right)$ where P is the pressure difference between the ends of the vessel and q is the viscosity of the blood. Find the average velocity (with respect to r) over the interval $0 \leq b \leq B \text {. }$

Find the average value of the function on the given interval. $h(x)=7 e^{\sin x} \cos x,\left[0, \frac{\pi}{2}\right]$

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=4 x, y=4 x e^{1-x / 2}, \text { about } y=16$

Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth. $y=0, y=-x^{4}+6 x^{3}-x^{2}+6 x$

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate x-coordinates of the points of intersection of the graphs. Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the x-axis. Round answers to two decimal places. y = $\frac{1}{5}$ $x^{5}$ , y = 3 $x^{2}$ - $x^{3}$

An aquarium 6 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the facts that the density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3} \text { and } g \approx 9.8 \text {. }$ )

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=\ln x, y=1, y=3, x=0 \text {; about the } y \text { - axis }$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 4 - $x^{2}$ , y = 0; the line y = 5