AP Calculus BC Unit 8 Practice Test & Exam | 33 Questions

1. [Skill: 1.E | Topic: 8.1] The function $f$ is defined by $f(x) = 3x^2 - 6x$. What is the average value of $f$ on the closed interval $[2, 5]$?

(A)18(B)54(C)27(D)6

2. [Skill: 2.B | Topic: 8.1] The graph of the function $g$ on the interval $[0, 8]$ consists of two line segments connecting the points $(0, 0)$, $(4, 8)$, and $(8, 0)$. What is the average value of $g$ on the interval $[0, 8]$?

(A)4(B)32(C)8(D)2

3. [Skill: 3.D | Topic: 8.1] The rate at which water flows into a tank, in gallons per minute, is modeled by the function $R(t) = 20 + 5\sin(\frac{\pi t}{12})$ for $0 \le t \le 12$ minutes. What is the average rate of water flow, in gallons per minute, over the 12-minute interval?

(A)20 + $\frac{10}{\pi}$(B)20(C)240 + $\frac{120}{\pi}$(D)20 + $\frac{5}{\pi}$

4. Let $R$ be the region in the first quadrant bounded by the graph of $x = y^2$, the line $x = 4$, and the x-axis. Which of the following definite integrals gives the volume of the solid generated when $R$ is revolved around the vertical line $x = 4$?

(A)$\pi \int_{0}^{2} (4 - y^2)^2 \, dy$(B)$\pi \int_{0}^{2} (y^2 - 4)^2 \, dy$(C)$\pi \int_{0}^{4} (4 - \sqrt{x})^2 \, dx$(D)$\pi \int_{0}^{4} (\sqrt{x} - 4)^2 \, dx$

5. Let $R$ be the region bounded by the graph of $y = e^x$, the vertical line $x = 1$, and the horizontal line $y = 1$. Which of the following definite integrals gives the volume of the solid generated when $R$ is revolved about the horizontal line $y = 1$?

(A)$\pi \int_{0}^{1} (e^x)^2 \, dx$(B)$\pi \int_{0}^{1} (e^x - 1)^2 \, dx$(C)$\pi \int_{1}^{e} (\ln y - 1)^2 \, dy$(D)$\pi \int_{1}^{e} (1 - \ln y)^2 \, dy$

6. Let $R$ be the region enclosed by the graphs of $y = \sqrt{x}$ and $y = x$. The volume of the solid generated by revolving $R$ about the horizontal line $y = -1$ is given by which of the following integrals?

(A)$\pi \int_{0}^{1} ((\sqrt{x})^2 - (x)^2) \, dx$(B)$\pi \int_{0}^{1} ((\sqrt{x} - 1)^2 - (x - 1)^2) \, dx$(C)$\pi \int_{0}^{1} ((\sqrt{x} + 1)^2 - (x + 1)^2) \, dx$(D)$\pi \int_{0}^{1} (\sqrt{x} - x)^2 \, dx$

7. Let $R$ be the region in the first quadrant bounded by the graphs of $y = 2\sqrt{x}$ and $y = x$. Which of the following definite integrals gives the volume of the solid generated when $R$ is revolved around the $x$-axis?

(A)$\displaystyle \pi \int_0^4 $(2$\sqrt{x} $- x)^2 $\, $dx(B)$\displaystyle \pi \int_0^4 $(4x - x^2) $\, $dx(C)$\displaystyle \pi $ $\int_0^4 $(x^2 - 4x) $\, $dx(D)$\displaystyle \pi \int_0^2 $(4y^2 - y^4) $\, $dy

8. Let $R$ be the region bounded by the graph of $y = e^x$, the horizontal line $y = 1$, and the vertical line $x = 2$. Which of the following integrals represents the volume of the solid generated when $R$ is revolved around the horizontal line $y = -1$?

(A)$\displaystyle \pi \int_0^2 $(e^{2x} - 1) $\, $dx(B)$\displaystyle \pi \int_0^2 $((e^x - 1)^2 - 2^2) $\, $dx(C)$\displaystyle \pi \int_0^2 $((e^x + 1)^2 - 4) $\, $dx(D)$\displaystyle \pi \int_0^2 $(e^x + 1 - 2)^2 $\, $dx

9. Let $R$ be the region in the first quadrant bounded by the graphs of $y = x^3$ and $y = 4x$. Which of the following expressions gives the volume of the solid generated when $R$ is revolved around the $y$-axis?

(A)$\displaystyle \pi \int_0^2 $((4x)^2 - (x^3)^2) $\, $dx(B)$\displaystyle \pi \int_0^8 \left$( y^{2/3} - $\frac{y^2}{16} \right$) $\, $dy(C)$\displaystyle \pi \int_0^8 \left$( $\frac{y^2}{16} $- y^{2/3} $\right$) $\, $dy(D)$\displaystyle \pi \int_0^8 \left$( y^{1/3} - $\frac{y}{4} \right$)^2 $\, $dy

10. [Skill: 1.E | Topic: 8.12] Let $R$ be the region in the first quadrant bounded by the graphs of $y = \sqrt{x}$ and $y = x^2$. Which of the following definite integrals represents the volume of the solid generated when $R$ is revolved about the horizontal line $y = -1$?

(A)$\pi \int_{0}^{1} \left[ (\sqrt{x} + 1)^2 - (x^2 + 1)^2 \right] \, dx$(B)$\pi \int_{0}^{1} \left[ (\sqrt{x} - 1)^2 - (x^2 - 1)^2 \right] \, dx$(C)$\pi \int_{0}^{1} \left[ (x^2 + 1)^2 - (\sqrt{x} + 1)^2 \right] \, dx$(D)$\pi \int_{0}^{1} \left[ (\sqrt{x})^2 - (x^2)^2 \right] \, dx$

11. [Skill: 1.E | Topic: 8.12] Let $R$ be the region enclosed by the graphs of $y = \frac{4}{x}$, $x = 1$, and $y = 2$. Which of the following is an expression for the volume of the solid generated when $R$ is revolved about the vertical line $x = 3$?

(A)$\pi \int_{2}^{4} \left[ \left(3 - \frac{4}{y}\right)^2 - (2)^2 \right] \, dy$(B)$\pi \int_{2}^{4} \left[ (2)^2 - \left(3 - \frac{4}{y}\right)^2 \right] \, dy$(C)$\pi \int_{1}^{2} \left[ \left(3 - \frac{4}{x}\right)^2 - (2)^2 \right] \, dx$(D)$\pi \int_{1}^{2} \left[ (2)^2 - \left(3 - \frac{4}{x}\right)^2 \right] \, dx$

12. [Skill: 2.B | Topic: 8.12] Let $R$ be the region in the first quadrant bounded by the graph of $y = \cos x$, the coordinate axes, and the line $x = \frac{\pi}{2}$. Which of the following integrals gives the volume of the solid generated by revolving $R$ about the line $y = 2$?

(A)$\pi \int_{0}^{\frac{\pi}{2}} \left[ (2 - \cos x)^2 - (2)^2 \right] \, dx$(B)$\pi \int_{0}^{\frac{\pi}{2}} \left[ (2)^2 - (2 - \cos x)^2 \right] \, dx$(C)$\pi \int_{0}^{\frac{\pi}{2}} \left[ (\cos x + 2)^2 - (2)^2 \right] \, dx$(D)$\pi \int_{0}^{\frac{\pi}{2}} \left[ (\cos x - 2)^2 \right] \, dx$

13. [Skill: 1.E | Topic: 8.13] Which of the following definite integrals represents the length of the curve defined by the function $y = \frac{x^3}{3} + \frac{1}{4x}$ on the interval $[1, 3]$?

(A)$\int_1^3 \left( x^2 + \frac{1}{4x^2} \right) dx$(B)$\int_1^3 \left( x^2 - \frac{1}{4x^2} \right) dx$(C)$\int_1^3 \sqrt{x^4 + \frac{1}{16x^4}} \, dx$(D)$\int_1^3 \left( x^2 + \frac{1}{2} + \frac{1}{4x^2} ight) dx$

14. [Skill: 1.E | Topic: 8.13] Let $g$ be the function defined by $g(x) = \int_0^x \sqrt{\cos(2t)} \, dt$ for $0 \le x \le \frac{\pi}{4}$. What is the length of the graph of $g$ from $x=0$ to $x=\frac{\pi}{4}$?

(A)1(B)$\sqrt{2}$(C)$\frac{\sqrt{2}}{2}$(D)$\frac{\pi}{4}$

Refer to the figure below.

15. A particle moves along the x-axis such that its velocity v(t) at time t, for 0 ≤ t ≤ 6, is given by the graph shown above. The graph consists of a triangle with vertices at (0, 0), (2, 4), and (4, 0), and a semi-circle centered at (5, 0) with a radius of 1 that lies below the t-axis. If the position of the particle at time t=0 is x(0) = 5, what is the position of the particle at time t=6?

(A)13 - π/2(B)13 + π/2(C)8 - π/2(D)5 + 8 + π/2

16. A particle moves along the x-axis with velocity given by v(t) = 3t^2 - 6t for time t ≥ 0. If the particle is at position x = 2 at time t = 0, what is the total distance traveled by the particle from t = 0 to t = 3?

(A)0(B)2(C)4(D)8

17. A storage tank initially contains 500 gallons of water. Water is pumped into the tank at a rate of $R(t)$ gallons per minute, where $t$ is the time in minutes since pumping began. Simultaneously, water leaks out of the tank at a constant rate of 2 gallons per minute. Which of the following expressions gives the amount of water in the tank, in gallons, at time $t = 20$ minutes?

(A)$500 + \int_0^{20} (R(t) - 2) \, dt$(B)$500 + \int_0^{20} R(t) \, dt - 2$(C)$\int_0^{20} (R(t) - 2) \, dt$(D)$500 + R(20) - 40$

18. The rate at which a pollutant is removed from a lake is given by the differentiable function $R(t)$, measured in kilograms per day, where $t$ is the number of days since the cleanup process began. If $\int_0^{10} R(t) \, dt = 450$, which of the following best interprets this mathematical statement in the context of the problem?

(A)The average rate at which the pollutant was removed over the first 10 days was 450 kilograms per day.(B)The amount of pollutant in the lake decreased by 450 kilograms during the first 10 days.(C)At $t=10$ days, the pollutant is being removed at a rate of 450 kilograms per day.(D)There are 450 kilograms of pollutant remaining in the lake after 10 days.

19. Let $R$ be the region bounded by the graphs of $y = x^2 - 4$ and $y = x - 2$. Which of the following definite integrals represents the area of region $R$?

(A)$\int_{-2}^{1} \left( (x^2 - 4) - (x - 2) \right) \, dx$(B)$\int_{-1}^{2} \left( (x - 2) - (x^2 - 4) \right) \, dx$(C)$\int_{-1}^{2} \left( (x^2 - 4) - (x - 2) \right) \, dx$(D)$\int_{-2}^{1} \left( (x - 2) - (x^2 - 4) \right) \, dx$

20. The functions $f$ and $g$ are continuous on the closed interval $[a, c]$. The graphs of $f$ and $g$ intersect at $x=a$, $x=b$, and $x=c$, where $a < b < c$. If $f(x) \ge g(x)$ for $a \le x \le b$ and $g(x) \ge f(x)$ for $b \le x \le c$, which of the following expressions represents the area of the region enclosed by the graphs of $f$ and $g$ between $x=a$ and $x=c$?

(A)$\int_{a}^{c} (f(x) - g(x)) \, dx$(B)$\int_{a}^{c} |f(x) + g(x)| \, dx$(C)$\int_{a}^{b} (f(x) - g(x)) \, dx + \int_{b}^{c} (g(x) - f(x)) \, dx$(D)$\int_{a}^{b} (g(x) - f(x)) \, dx + \int_{b}^{c} (f(x) - g(x)) \, dx$

21. Let $R$ be the region in the first quadrant bounded by the graph of $x = y^3$ and the line $x = 4y$. Which of the following definite integrals represents the area of region $R$?

(A)$\int_{0}^{2} (y^3 - 4y) \, dy$(B)$\int_{0}^{2} (4y - y^3) \, dy$(C)$\int_{0}^{8} (4y - y^3) \, dy$(D)$\int_{0}^{8} (y^3 - 4y) \, dy$

22. The region $R$ is bounded by the graph of $x = e^y$, the horizontal line $y = 1$, and the vertical line $x = 1$. What is the area of region $R$?

(A)$e - 2$(B)$e - 1$(C)$e$(D)$\frac{1}{2}e^2 - \frac{3}{2}$

23. Which of the following expressions represents the total area of the region enclosed by the graphs of $f(x) = x^3$ and $g(x) = 4x$?

(A)$\int_{-2}^{2} (4x - x^3) \, dx$(B)$\int_{-2}^{0} (x^3 - 4x) \, dx + \int_{0}^{2} (4x - x^3) \, dx$(C)$\int_{-2}^{0} (4x - x^3) \, dx + \int_{0}^{2} (x^3 - 4x) \, dx$(D)$\int_{-2}^{2} (x^3 - 4x) \, dx$

24. Let $R$ be the region enclosed by the graphs of $y = 2x$ and $y = x^2$. The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?

(A)$\frac{16}{15}$(B)$\frac{4}{3}$(C)$\frac{64}{15}$(D)$\frac{16\pi}{15}$

25. The base of a solid is the region in the first quadrant bounded by the graph of $y = \ln x$, the vertical line $x = e$, and the $x$-axis. For this solid, each cross section perpendicular to the $y$-axis is a rectangle whose height is 3 times the length of its base in the $xy$-plane. Which of the following integrals gives the volume of the solid?

(A)$\int_{1}^{e} $3(e - x)^2 $\, $dx(B)$\int_{0}^{1} $3(e - e^y)^2 $\, $dy(C)$\int_{0}^{1} $(e - e^y)^2 $\, $dy(D)$\pi \int_{0}^{1} $(e^2 - e^{2y}) $\, $dy

26. Let $R$ be the region in the first quadrant bounded by the graph of $y = \sqrt{4x}$ and the line $y = x$. The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are semicircles with diameters in $R$. Which of the following definite integrals gives the volume of the solid?

(A)$\frac{\pi}{2} \int_{0}^{4} $($\sqrt{4x} $- x)^2 $\, $dx(B)$\frac{\pi}{8} \int_{0}^{4} $($\sqrt{4x} $- x)^2 $\, $dx(C)$\frac{\pi}{2} \int_{0}^{4} $(4x - x^2) $\, $dx(D)$\frac{\pi}{8} \int_{0}^{4} $($\sqrt{4x} $- x) $\, $dx

27. Let $R$ be the region enclosed by the graphs of $y = \ln x$, the line $x = e$, and the $x$-axis. The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is an equilateral triangle. Which of the following expressions represents the volume of the solid?

(A)$\int_{1}^{e} \frac{\sqrt{3}}{4} $($\ln $x)^2 $\, $dx(B)$\int_{1}^{e} \frac{1}{2} $($\ln $x)^2 $\, $dx(C)$\int_{0}^{e} \frac{\sqrt{3}}{4} $($\ln $x)^2 $\, $dx(D)$\int_{1}^{e} \sqrt{3} $($\ln $x) $\, $dx

28. Let $R$ be the region in the first quadrant bounded by the graph of $y = e^{x/2}$, the vertical line $x = 2$, and the coordinate axes. What is the volume of the solid generated when $R$ is revolved about the $x$-axis?

(A)2$\pi$(e - 1)(B)$\pi$(e^2 - 1)(C)$\pi$(e^2 - 2)(D)$\frac{\pi}{2}$(e^2 - 1)

29. Let $R$ be the region in the first quadrant bounded by the graph of $y = x^3$, the horizontal line $y = 8$, and the $y$-axis. Which of the following definite integrals represents the volume of the solid generated when $R$ is revolved about the $y$-axis?

(A)$\pi \int_{0}^{2} $(8 - x^3)^2 $\, $dx(B)$\pi \int_{0}^{2} $x^6 $\, $dx(C)$\pi \int_{0}^{8} $y^{1/3} $\, $dy(D)$\pi \int_{0}^{8} $y^{2/3} $\, $dy

30. Let $R$ be the region enclosed by the graphs of $f(x) = x^3 - 2x^2$ and $g(x) = -x^2 + 2x$. Which of the following expressions gives the total area of region $R$?

(A)$\displaystyle \int_{-1}^{2} \left( (x^3 - 2x^2) - (-x^2 + 2x) \right) \, dx$(B)$\displaystyle \int_{-1}^{0} \left( (x^3 - 2x^2) - (-x^2 + 2x) \right) \, dx + \int_{0}^{2} \left( (-x^2 + 2x) - (x^3 - 2x^2) \right) \, dx$(C)$\displaystyle \int_{-1}^{0} \left( (-x^2 + 2x) - (x^3 - 2x^2) \right) \, dx + \int_{0}^{2} \left( (x^3 - 2x^2) - (-x^2 + 2x) \right) \, dx$(D)$\displaystyle \int_{0}^{2} \left( (-x^2 + 2x) - (x^3 - 2x^2) \right) \, dx$

AP Calculus BC Unit 8: Applications of Integration Practice Exam

Exam Details

A. Multiple-Choice Questions (30 Questions)

B. Short Answer Questions (3 Questions)

AP Calculus BC Unit 8: Applications of Integration Practice Exam

Exam Details

A. Multiple-Choice Questions (30 Questions)

B. Short Answer Questions (3 Questions)