AP Calculus AB Unit 8: Applications of Integration

The average value of a function, position, displacement and total distance from velocity, accumulation in context with an initial amount, area between curves in $x$ and in $y$ (including curves that cross more than once), volumes with square, rectangle, triangle and semicircle cross sections, and volumes of revolution by discs and washers about the axes and about other horizontal and vertical lines.

CED topics in this unit: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.10, 8.11, 8.12.

Key formulas and facts

• Average value: $f_{\text{avg}}=\frac{1}{b-a}\int_a^bf(x)\,dx$
• Displacement $=\int_a^bv(t)\,dt$; total distance $=\int_a^b|v(t)|\,dt$; $x(b)=x(a)+\int_a^bv(t)\,dt$
• Area between curves: $\int_a^b(\text{top}-\text{bottom})\,dx$ or $\int_c^d(\text{right}-\text{left})\,dy$
• Cross sections: $V=\int_a^bA(x)\,dx$; square $A=s^2$; semicircle on diameter $s$: $A=\frac{\pi}{8}s^2$; equilateral triangle $A=\frac{\sqrt3}{4}s^2$
• Disc: $V=\pi\int_a^bR(x)^2\,dx$; washer: $V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)dx$
• About $y=k$: radius $=|f(x)-k|$; about $x=h$: integrate in $y$ with radius $|g(y)-h|$

Worked examples

• Average value of $\sin x$ on $[0,\pi]$: $\frac1\pi\int_0^\pi\sin x\,dx=\frac2\pi$
• Position: $v(t)=2t-4$ and $x(0)=3$ give $x(5)=3+\int_0^5(2t-4)\,dt=8$; the distance traveled is $\int_0^5|2t-4|\,dt=13$
• Area: $y=x$ and $y=x^2$ meet at $0$ and $1$, so the area is $\int_0^1(x-x^2)\,dx=\frac16$
• Washer about $y=-1$ for the region under $y=\sqrt x$ on $[0,4]$: $V=\pi\int_0^4\left((\sqrt x+1)^2-1^2\right)dx=\frac{56\pi}{3}$
• Square cross sections on the region under $y=4-x^2$: $V=\int_{-2}^{2}(4-x^2)^2\,dx=\frac{512}{15}$

Common traps

• Washers: squaring the difference, $(R-r)^2$, instead of using $R^2-r^2$.
• Revolving about $y=k$ but using $f(x)$ as the radius instead of $|f(x)-k|$.
• Giving the displacement when the question asks for total distance.
• Integrating in $x$ when the region's left and right boundaries are functions of $y$.
• Forgetting the initial value: $x(b)=x(a)+\int_a^bv(t)\,dt$, not just the integral.

Practice questions

78 multiple-choice and 6 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

What is the average value of $f(x)=3x^2-2x$ on the interval $[1,4]$?

1. $48$
2. $\frac{41}{2}$
3. $16$
4. $12$

Question 2 · Multiple choice

A particle moves along the $x$-axis with velocity $v(t)=\ln(t^2+1)-1$ for $0\le t\le3$. What is the total distance the particle travels over this interval?

1. $0.406$
2. $1.189$
3. $1.973$
4. $0.783$

Question 3 · Multiple choice

A tank holds $50$ gallons of water at $t=0$. Water flows in at $E(t)=8+3\sqrt t$ gallons per minute and out at $10$ gallons per minute. How many gallons are in the tank at $t=4$ minutes?

1. $58$
2. $8$
3. $66$
4. $42$

Question 4 · Multiple choice

Find the area enclosed between the graphs of $y=6-x^2$ and $y=x$.

1. $\frac{22}{3}$
2. $\frac{27}{2}$
3. $\frac{125}{3}$
4. $\frac{125}{6}$

Question 5 · Multiple choice

What is the area of the region bounded by $x=y^2$ and $x=2y+3$?

1. $\frac{16}{3}$
2. $\frac{32}{3}$
3. $9$
4. $\frac{64}{3}$

Question 6 · Multiple choice

What is the total area of the regions bounded by $y=x^3$ and $y=4x$?

1. $0$
2. $4$
3. $16$
4. $8$

Question 7 · Multiple choice

The base of a solid is the region bounded by $y=\sqrt{x}$, the $x$-axis and $x=4$. Cross sections perpendicular to the $x$-axis are semicircles whose diameters lie in the base. What is the volume of the solid?

1. $4 \pi$
2. $\pi$
3. $2 \pi$
4. $8$

Question 8 · Multiple choice

The region bounded by $y=x$ and $y=x^2$ is revolved about the line $y=2$. What is the volume of the solid?

1. $\frac{2 \pi}{15}$
2. $\frac{7 \pi}{15}$
3. $\frac{8 \pi}{15}$
4. $\frac{\pi}{30}$

Question 9 · Multiple choice

The region in the first quadrant bounded by $y=x^2$, $y=4$ and the $y$-axis is revolved about the $y$-axis. What is the volume of the solid?

1. $16 \pi$
2. $8 \pi$
3. $\frac{128 \pi}{5}$
4. $4 \pi$

Question 10 · Free response (9 points)

Let $R$ be the region in the first quadrant bounded by the graphs of $y=5-x^2$ and $y=2^x$ and the $y$-axis.

1. (a) Find the area of $R$. [2 points]
2. (b) Find the volume of the solid generated when $R$ is revolved about the horizontal line $y=-1$. [3 points]
3. (c) $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Find the volume of the solid. [2 points]
4. (d) The vertical line $x=k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives $k$. [2 points]

Question 11 · Free response (9 points)

A particle moves along the $x$-axis with velocity $v(t)=t^2-9t+18$ for $0\le t\le8$. At $t=0$ the particle is at $x=2$.

1. (a) Find the position of the particle at $t=6$. [2 points]
2. (b) Find the total distance traveled by the particle for $0\le t\le8$. [3 points]
3. (c) Find the average velocity of the particle for $0\le t\le8$. [1 point]
4. (d) Find the position of the particle when it is farthest to the right for $0\le t\le8$. Justify your answer. [3 points]

Question 12 · Multiple choice

What is the average value of $f(x)=x^2$ on the interval $[0,3]$?

1. $9$
2. $\frac{9}{2}$
3. $3$
4. $1$

Question 13 · Multiple choice

What is the average value of $\sin x$ on the interval $[0,\pi]$?

1. $\frac{2}{\pi}$
2. $2$
3. $\frac{1}{2}$
4. $\frac{1}{\pi}$

Question 14 · Multiple choice

What is the average value of $f(x)=\dfrac1x$ on the interval $[1,e]$?

1. $\frac{1}{-1 + e}$
2. $1$
3. $-1 + e$
4. $e^{-1}$

Question 15 · Multiple choice

The temperature in a greenhouse is $T(t)=60+10\sin\!\left(\frac{\pi t}{12}\right)$ degrees Fahrenheit, $t$ hours after 6 a.m. What is the average temperature over $0\le t\le8$?

1. $68.660$
2. $70.000$
3. $60.000$
4. $67.162$

Question 16 · Multiple choice

The velocity of a particle moving on a line is $v(t)=t^2-9$, for $0\le t\le4$. What is its displacement over this time interval?

1. $\frac{64}{3}$
2. $- \frac{44}{3}$
3. $18$
4. $\frac{44}{3}$

Question 17 · Multiple choice

The velocity of a particle moving on a line is $v(t)=t^2-9$, for $0\le t\le4$. What total distance does it travel over this time interval?

1. $\frac{64}{3}$
2. $- \frac{44}{3}$
3. $18$
4. $\frac{10}{3}$

Question 18 · Multiple choice

A particle has velocity $v(t)=3t^2-2t$ and position $x(0)=5$. What is $x(2)$?

1. $9$
2. $4$
3. $13$
4. $10$

Question 19 · Multiple choice

A particle moves along a line with velocity $v(t)=\cos(t^2)$ for $0\le t\le2$. What is the total distance traveled?

1. $0.461$
2. $1.493$
3. $1.032$
4. $2.000$

Question 20 · Multiple choice

Water flows into a tank at $50+10t$ liters per minute and drains out at $30+5t$ liters per minute. At $t=0$ the tank holds $100$ liters. How many liters are in the tank at $t=4$?

1. $180$
2. $120$
3. $400$
4. $220$

Question 21 · Multiple choice

Water enters a reservoir at the rate $E(t)=20\sin\!\left(\frac t2\right)+30$ million gallons per day and leaves at the constant rate $25$ million gallons per day, for $0\le t\le6$ days. What is the net change in the amount of water over these $6$ days, in millions of gallons?

1. $259.600$
2. $150.000$
3. $39.600$
4. $109.600$

Question 22 · Multiple choice

People enter a stadium at the rate $r(t)=1200te^{-t/2}$ people per hour, $t$ hours after the gates open. How many people enter during the first $6$ hours?

1. $3600.000$
2. $358.468$
3. $3844.088$
4. $4800.000$

Question 23 · Multiple choice

Grain is added to a silo at $40$ tons per hour and removed at $10t$ tons per hour, for $0\le t\le8$. At what time is the amount of grain in the silo greatest?

1. $8$
2. $4$
3. $0$
4. $2$

Question 24 · Multiple choice

What is the area of the region between the graphs of $y=x^2$ and $y=2x$?

1. $\frac{8}{3}$
2. $4$
3. $\frac{2}{3}$
4. $\frac{4}{3}$

Question 25 · Multiple choice

What is the area of the region between the graphs of $y=\sqrt x$ and $y=x^2$?

1. $\frac{2}{3}$
2. $1$
3. $\frac{1}{3}$
4. $\frac{1}{6}$

Question 26 · Multiple choice

What is the area of the region bounded by $y=e^x$, $y=1$ and $x=2$?

1. $-3 + e^{2}$
2. $-1 + e^{2}$
3. $-2 + e^{2}$
4. $e^{2}$

Question 27 · Multiple choice

Find the area enclosed by the graphs of $y=6-x^2$ and $y=x$.

1. $\frac{22}{3}$
2. $\frac{27}{2}$
3. $\frac{125}{3}$
4. $\frac{125}{6}$

Question 28 · Multiple choice

What is the area of the region bounded by $y=\cos x$ and $y=x^2$?

1. $0.824$
2. $1.095$
3. $0.547$
4. $1.648$

Question 29 · Multiple choice

What is the area of the region bounded by $x=y^2$ and $x=4$?

1. $\frac{16}{3}$
2. $8$
3. $\frac{64}{3}$
4. $\frac{32}{3}$

Question 30 · Multiple choice

What is the area of the region bounded by the curve $x=2y-y^2$ and the $y$-axis?

1. $\frac{2}{3}$
2. $\frac{4}{3}$
3. $\frac{8}{3}$
4. $1$

Question 31 · Multiple choice

The region $R$ is bounded by $y=\sqrt{x}$, $y=2$ and the $y$-axis. Which integral in terms of $y$ gives the area of $R$?

1. $\displaystyle\int_0^2(2-y^2)\,dy$
2. $\displaystyle\int_0^2y^2\,dy$
3. $\displaystyle\int_0^4y^2\,dy$
4. $\displaystyle\int_0^2\sqrt{y}\,dy$

Question 32 · Multiple choice

What is the area of the region between $x=y^3$ and $x=y$ for $0\le y\le1$?

1. $\frac{1}{2}$
2. $\frac{1}{4}$
3. $\frac{3}{4}$
4. $\frac{1}{12}$

Question 33 · Multiple choice

What is the area of the region bounded by $x=\sin y$ and $x=0$ for $0\le y\le\pi$?

1. $1$
2. $\pi$
3. $\frac{\pi}{2}$
4. $2$

Question 34 · Multiple choice

What is the area of the region bounded by $x=y^2-4$ and $x=2-y$?

1. $\frac{27}{2}$
2. $\frac{61}{6}$
3. $\frac{125}{6}$
4. $\frac{125}{3}$

Question 35 · Multiple choice

What is the total area of the regions between the graphs of $y=x^3$ and $y=x$?

1. $0$
2. $\frac{1}{4}$
3. $1$
4. $\frac{1}{2}$

Question 36 · Multiple choice

Find the combined area of the regions enclosed by the $x$-axis and the graph of $y=x^3-9x$.

1. $0$
2. $\frac{81}{4}$
3. $81$
4. $\frac{81}{2}$

Question 37 · Multiple choice

What is the total area between the graphs of $y=\sin x$ and $y=\cos x$ for $0\le x\le\pi$?

1. $2$
2. $2 \sqrt{2}$
3. $\sqrt{2}$
4. $-2 + 2 \sqrt{2}$

Question 38 · Multiple choice

What is the total area of the regions between the graphs of $y=x^3-2x^2$ and $y=3x$?

1. $\frac{45}{4}$
2. $\frac{7}{12}$
3. $\frac{32}{3}$
4. $\frac{71}{6}$

Question 39 · Multiple choice

What is the total area of the regions between the graphs of $y=x\sin x$ and $y=\dfrac x2$ for $0\le x\le\pi$?

1. $0.523$
2. $1.477$
3. $2.467$
4. $1.000$

Question 40 · Multiple choice

The curve $y=x^4-4x^2$ dips below the $x$-axis between its zeros. What is the total area enclosed by the curve and the axis?

1. $\frac{128}{15}$
2. $\frac{64}{15}$
3. $0$
4. $\frac{128}{3}$

Question 41 · Multiple choice

What is the total area of the regions between the graphs of $y=x^3-2x$ and $y=\sin x$?

1. $1.952$
2. $3.903$
3. $1.618$
4. $0.000$

Question 42 · Multiple choice

The base of a solid is the region bounded by $y=\sqrt x$, the $x$-axis and $x=4$. Cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?

1. $8$
2. $\frac{16}{3}$
3. $8 \pi$
4. $16$

Question 43 · Multiple choice

A solid's base is the region enclosed by $y=9-x^2$ and the $x$-axis. Every cross section perpendicular to the $x$-axis is a square. Find the volume of the solid.

1. $36$
2. $\frac{1296}{5}$
3. $\frac{648}{5}$
4. $\frac{1296 \pi}{5}$

Question 44 · Multiple choice

The base of a solid is the region between $y=x$ and $y=x^2$. Cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?

1. $\frac{1}{6}$
2. $\frac{1}{30}$
3. $\frac{2}{15}$
4. $\frac{1}{15}$

Question 45 · Multiple choice

The base of a solid is the region under $y=\dfrac1x$ above the $x$-axis for $1\le x\le3$. Cross sections perpendicular to the $x$-axis are rectangles whose height is twice the length of the base. What is the volume?

1. $\frac{2}{3}$
2. $2 \ln{\left(3 \right)}$
3. $\frac{4}{3}$
4. $\frac{8}{3}$

Question 46 · Multiple choice

The base of a solid is the region bounded by $x=y^2$, $x=4$ and the $x$-axis, in the first quadrant. Cross sections perpendicular to the $y$-axis are squares. What is the volume?

1. $\frac{16}{3}$
2. $\frac{256}{15}$
3. $\frac{512}{15}$
4. $\frac{128}{15}$

Question 47 · Multiple choice

The base of a solid is the region under $y=\sin x$ for $0\le x\le\pi$. Cross sections perpendicular to the $x$-axis are squares. What is the volume?

1. $2$
2. $\frac{\pi}{2}$
3. $\pi$
4. $1$

Question 48 · Multiple choice

The base of a solid is the region under $y=\sqrt x$ for $0\le x\le4$. Cross sections perpendicular to the $x$-axis are semicircles with diameters in the base. What is the volume?

1. $2 \pi$
2. $\pi$
3. $4 \pi$
4. $\frac{\pi}{2}$

Question 49 · Multiple choice

The base of a solid is the region between $y=x$ and the $x$-axis for $0\le x\le3$. Cross sections perpendicular to the $x$-axis are equilateral triangles. What is the volume?

1. $\frac{9 \sqrt{3}}{4}$
2. $\frac{9}{2}$
3. $\frac{9 \sqrt{3}}{2}$
4. $\frac{3 \sqrt{3}}{4}$

Question 50 · Multiple choice

The base of a solid is the triangle bounded by $y=2-x$ and the coordinate axes. Cross sections perpendicular to the $x$-axis are isosceles right triangles with one leg in the base. What is the volume?

1. $\frac{8}{3}$
2. $\frac{2}{3}$
3. $\frac{4}{3}$
4. $2$

Question 51 · Multiple choice

The base of a solid is the region between $y=4$ and $y=x^2$. Cross sections perpendicular to the $x$-axis are semicircles with diameters in the base. What is the volume?

1. $\frac{128 \pi}{15}$
2. $\frac{64 \pi}{15}$
3. $\frac{512 \pi}{15}$
4. $\frac{32 \pi}{15}$

Question 52 · Multiple choice

The base of a solid is the triangle bounded by $x+y=2$ and the coordinate axes. Cross sections perpendicular to the $y$-axis are equilateral triangles. What is the volume?

1. $\frac{2 \sqrt{3}}{3}$
2. $\frac{4 \sqrt{3}}{3}$
3. $\frac{\sqrt{3}}{3}$
4. $\frac{8}{3}$

Question 53 · Multiple choice

The base of a solid is the region bounded by $y=\ln x$, the $x$-axis and $x=e$. Cross sections perpendicular to the $x$-axis are semicircles with diameters in the base. What is the volume?

1. $0.564$
2. $0.393$
3. $1.128$
4. $0.282$

Question 54 · Multiple choice

The region under $y=\sqrt x$ for $0\le x\le4$ is revolved about the $x$-axis. What is the volume of the solid?

1. $\frac{16 \pi}{3}$
2. $4 \pi$
3. $16 \pi$
4. $8 \pi$

Question 55 · Multiple choice

The region under $y=x^2$ for $0\le x\le2$ is revolved about the $x$-axis. What is the volume?

1. $\frac{8 \pi}{3}$
2. $\frac{16 \pi}{5}$
3. $\frac{32 \pi}{5}$
4. $\frac{32 \pi}{3}$

Question 56 · Multiple choice

The region under $y=e^x$ for $0\le x\le1$ is revolved about the $x$-axis. What is the volume?

1. $\pi \left(-1 + e\right)$
2. $\pi \left(-1 + e^{2}\right)$
3. $\frac{\pi \left(-1 + e^{2}\right)}{2}$
4. $\frac{\pi e^{2}}{2}$

Question 57 · Multiple choice

The region bounded by $y=x^3$, $y=8$ and the $y$-axis is revolved about the $y$-axis. What is the volume?

1. $\frac{96 \pi}{5}$
2. $12 \pi$
3. $\frac{48 \pi}{5}$
4. $64 \pi$

Question 58 · Multiple choice

The region under $y=\dfrac1x$ for $1\le x\le3$ is revolved about the $x$-axis. What is the volume?

1. $\pi \ln{\left(3 \right)}$
2. $\frac{\pi}{3}$
3. $\frac{4 \pi}{3}$
4. $\frac{2 \pi}{3}$

Question 59 · Multiple choice

The region under $y=\sec x$ for $0\le x\le\frac\pi4$ is revolved about the $x$-axis. What is the volume?

1. $\frac{\pi^{2}}{4}$
2. $\sqrt{2} \pi$
3. $\frac{\pi}{2}$
4. $\pi$

Question 60 · Multiple choice

The region under $y=xe^{-x}$ for $0\le x\le2$ is revolved about the $x$-axis. What is the volume?

1. $0.190$
2. $1.880$
3. $0.299$
4. $0.598$

Question 61 · Multiple choice

The region between $y=x^2$ and $y=4$ is revolved about the line $y=4$. What is the volume?

1. $\frac{256 \pi}{5}$
2. $\frac{32 \pi}{3}$
3. $\frac{256 \pi}{15}$
4. $\frac{512 \pi}{15}$

Question 62 · Multiple choice

The region bounded by $y=\sqrt x$, $y=2$ and the $y$-axis is revolved about the line $y=2$. What is the volume?

1. $\frac{16 \pi}{3}$
2. $8 \pi$
3. $\frac{8 \pi}{3}$
4. $\frac{4 \pi}{3}$

Question 63 · Multiple choice

The region bounded by $x=y^2$ and $x=1$ is revolved about the line $x=1$. What is the volume?

1. $\frac{8 \pi}{15}$
2. $\frac{\pi}{2}$
3. $\frac{16 \pi}{15}$
4. $\frac{4 \pi}{3}$

Question 64 · Multiple choice

The region between $y=\cos x$ and $y=1$ for $0\le x\le\frac\pi2$ is revolved about the line $y=1$. What is the volume?

1. $\frac{\pi^{2}}{4}$
2. $- 2 \pi + \frac{3 \pi^{2}}{4}$
3. $\pi \left(-1 + \frac{\pi}{2}\right)$
4. $\frac{3 \pi^{2}}{4}$

Question 65 · Multiple choice

The region bounded by $y=x^3$, the $x$-axis and $x=1$ is revolved about the line $x=1$. What is the volume?

1. $\frac{\pi}{7}$
2. $\frac{\pi}{10}$
3. $\frac{\pi}{5}$
4. $\frac{3 \pi}{10}$

Question 66 · Multiple choice

The region bounded by $y=\ln x$, the $x$-axis and $x=e$ is revolved about the line $x=e$. What is the volume?

1. $3.902$
2. $1.242$
3. $7.804$
4. $2.257$

Question 67 · Multiple choice

The region between $y=x$ and $y=x^2$ is revolved about the $x$-axis. What is the volume?

1. $\frac{\pi}{6}$
2. $\frac{\pi}{30}$
3. $\frac{2 \pi}{15}$
4. $\frac{\pi}{15}$

Question 68 · Multiple choice

The region between $y=x$ and $y=x^2$ is revolved about the $y$-axis. What is the volume?

1. $\frac{2 \pi}{15}$
2. $\frac{\pi}{3}$
3. $\frac{\pi}{6}$
4. $\frac{\pi}{30}$

Question 69 · Multiple choice

The region between $y=\sqrt x$ and $y=\dfrac x2$ is revolved about the $x$-axis. What is the volume?

1. $\frac{4 \pi}{3}$
2. $\frac{16 \pi}{3}$
3. $8 \pi$
4. $\frac{8 \pi}{3}$

Question 70 · Multiple choice

The region between $y=x^2$ and $y=4$ is revolved about the $x$-axis. What is the volume?

1. $\frac{512 \pi}{15}$
2. $\frac{256 \pi}{5}$
3. $\frac{64 \pi}{5}$
4. $64 \pi$

Question 71 · Multiple choice

The region bounded by $y=e^x$, $y=1$ and $x=1$ is revolved about the $x$-axis. What is the volume?

1. $\frac{\pi \left(-1 + e^{2}\right)}{2}$
2. $\frac{\pi \left(-3 + e^{2}\right)}{2}$
3. $\pi \left(-2 + e\right)$
4. $\pi \left(-3 + e^{2}\right)$

Question 72 · Multiple choice

The region between $y=2x$ and $y=x^2$ is revolved about the $x$-axis. What is the volume?

1. $\frac{32 \pi}{15}$
2. $\frac{8 \pi}{3}$
3. $\frac{64 \pi}{15}$
4. $\frac{16 \pi}{15}$

Question 73 · Multiple choice

The region between $y=\sin x$ and $y=\dfrac x2$ in the first quadrant is revolved about the $x$-axis. What is the volume?

1. $0.531$
2. $1.669$
3. $0.839$
4. $3.338$

Question 74 · Multiple choice

The region between $y=x$ and $y=x^2$ is revolved about the line $y=-1$. What is the volume?

1. $\frac{2 \pi}{15}$
2. $\frac{7 \pi}{15}$
3. $\frac{8 \pi}{15}$
4. $\frac{\pi}{3}$

Question 75 · Multiple choice

The region between $y=x$ and $y=x^2$ is revolved about the line $y=2$. What is the volume?

1. $\frac{7 \pi}{15}$
2. $\frac{2 \pi}{15}$
3. $\frac{8 \pi}{15}$
4. $\frac{4 \pi}{5}$

Question 76 · Multiple choice

The region between $y=x^2$ and $y=4$ is revolved about the line $y=-1$. What is the volume?

1. $\frac{1088 \pi}{15}$
2. $\frac{256 \pi}{5}$
3. $\frac{832 \pi}{15}$
4. $\frac{1088 \pi}{5}$

Question 77 · Multiple choice

The region bounded by $y=e^x$, $y=1$ and $x=2$ is revolved about the line $y=-2$. Which integral gives the volume?

1. $\displaystyle\pi\int_0^2\left((e^x+2)^2-9\right)dx$
2. $\displaystyle\pi\int_0^2\left((e^x+2)^2-1\right)dx$
3. $\displaystyle\pi\int_0^2\left((e^x-2)^2-9\right)dx$
4. $\displaystyle\pi\int_0^2\left(e^x+2-3\right)^2dx$

Question 78 · Multiple choice

The region between $y=2\cos x$ and $y=1$ for $-\frac\pi3\le x\le\frac\pi3$ is revolved about the line $y=-1$. What is the volume?

1. $20.627$
2. $10.314$
3. $6.566$
4. $41.254$

Question 79 · Free response (9 points)

Let $R$ be the region between the graphs of $f(x)=4x-x^2$ and $g(x)=x$.

1. (a) Find the area of $R$. [2 points]
2. (b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis. [3 points]
3. (c) Find the volume of the solid generated when $R$ is revolved about the line $y=-2$. [2 points]
4. (d) The region $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Find the volume of the solid. [2 points]

Question 80 · Free response (9 points)

Let $S$ be the region in the plane enclosed by the $y$-axis and the parabola $x=9-y^2$.

1. (a) Find the area of $S$. [2 points]
2. (b) Find the volume of the solid generated when $S$ is revolved about the $y$-axis. [2 points]
3. (c) $S$ is the base of a solid whose cross sections perpendicular to the $y$-axis are semicircles with diameters in $S$. Find the volume. [2 points]
4. (d) Find the volume of the solid generated when $S$ is revolved about the line $x=-1$. [3 points]

Question 81 · Free response (9 points)

A particle moves along the $x$-axis with velocity $v(t)=t\cos t$ for $0\le t\le4$. At time $t=0$ the particle is at $x=2$.

1. (a) Find the acceleration of the particle at $t=2$. [1 point]
2. (b) At what time in $0<t<4$ does the particle change direction? Justify your answer. [2 points]
3. (c) Find the position of the particle at $t=4$. [2 points]
4. (d) Find the total distance traveled by the particle over $0\le t\le4$. [2 points]
5. (e) Find the average velocity of the particle over $0\le t\le4$. [2 points]

Question 82 · Free response (9 points)

Water flows into a tank at the rate $E(t)=30+10\sin\!\left(\frac t3\right)$ gallons per hour and drains out at the rate $L(t)=2t+25$ gallons per hour, for $0\le t\le12$ hours. At $t=0$ the tank holds $200$ gallons.

1. (a) Find the average rate at which water flows into the tank over $0\le t\le12$. [2 points]
2. (b) Is the amount of water in the tank increasing or decreasing at $t=6$? Give a reason. [2 points]
3. (c) How many gallons of water are in the tank at $t=12$? [2 points]
4. (d) At what time $t$, for $0\le t\le12$, is the amount of water in the tank greatest? Justify your answer. [3 points]

Question 83 · Multiple choice

Sand is added to a pile at the rate $S(t)=4+t$ tons per hour and removed at a constant $6$ tons per hour, for $0\le t\le6$. The pile holds $20$ tons at $t=0$. How many tons does it hold at $t=6$?

1. $38$
2. $26$
3. $20$
4. $62$

Question 84 · Multiple choice

The region bounded by $y=\sqrt x$, the $x$-axis and $x=4$ is revolved about the line $x=4$. What is the volume?

1. $\frac{512 \pi}{15}$
2. $8 \pi$
3. $\frac{128 \pi}{15}$
4. $\frac{256 \pi}{15}$