AP Calculus AB Unit 8: Applications of Integration Practice Exam

Assessment for Unit 8: Applications of Integration

Estimated Time: 60 minutes

Exam Details

Questions: 30 total (28 multiple choice, 2 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 8: Applications of Integration
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (28 Questions)

Select the one best answer for each question.

1. What is the average value of the function $f(x) = 6x^2 - 4x$ on the closed interval $[0, 3]$?

(A)12(B)21(C)36(D)42

2. The graph of the function $f$, defined on the interval $[0, 8]$, consists of two line segments connecting the points $(0,0)$ to $(4,6)$ and $(4,6)$ to $(8,0)$. What is the average value of $f$ on the interval $[0, 8]$?

(A)3(B)6(C)12(D)24

3. The velocity of a particle, $v(t)$, moving along a straight line is given in the table below for selected values of $t$. | $t$ (seconds) | 0 | 3 | 5 | | :--- | :--- | :--- | :--- | | $v(t)$ (m/s) | 10 | 20 | 25 | Using a Right Riemann sum with the two subintervals indicated by the table, what is the estimated average velocity of the particle on the interval $[0, 5]$?

(A)14 m/s(B)18.3 m/s(C)22 m/s(D)110 m/s

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

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

5. Let $R$ be the region bounded by the graph of $y = e^x$, the vertical lines $x = 0$ and $x = 1$, and the horizontal line $y = e$. Which of the following expressions gives the volume of the solid generated when $R$ is revolved about the line $y = e$?

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

6. Let $R$ be the region in the first quadrant bounded by the graph of $y = \frac{1}{x}$, the horizontal line $y = 4$, and the vertical line $x = 1$. Which of the following definite integrals gives the volume of the solid generated when $R$ is revolved about the line $y = 4$?

(A)$\pi \int_{1/4}^{1} (4 - \frac{1}{x})^2 \, dx$(B)$\pi \int_{1/4}^{1} (\frac{1}{x} - 4)^2 \, dx$(C)$\pi \int_{1/4}^{1} (4^2 - (\frac{1}{x})^2) \, dx$(D)$\pi \int_{1}^{4} (4 - \frac{1}{y})^2 \, dy$

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

(A)$\pi \int_{0}^{2} (4x - x^3)^2 \, dx$(B)$\pi \int_{0}^{2} \left( (4x)^2 - (x^3)^2 \right) \, dx$(C)$\pi \int_{0}^{2} \left( (x^3)^2 - (4x)^2 \right) \, dx$(D)$\pi \int_{0}^{8} \left( y^{2/3} - \frac{y^2}{16} \right) \, dy$

8. Let $R$ be the region bounded by the graph of $y = \ln x$, the vertical line $x = e$, and the $x$-axis. Which of the following expressions gives the volume of the solid generated when $R$ is revolved around the $y$-axis?

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

9. Let $R$ be the region in the first quadrant bounded by the graph of $y = x^2$, the line $y = 4$, and the $y$-axis. Which of the following integrals gives the volume of the solid generated when $R$ is revolved about the vertical line $x = 3$?

(A)$\pi \int_{0}^{4} \left( 9 - (3 - \sqrt{y})^2 \right) \, dy$(B)$\pi \int_{0}^{4} \left( (3 - \sqrt{y})^2 - 9 \right) \, dy$(C)$\pi \int_{0}^{2} \left( 9 - (3 - x^2)^2 \right) \, dx$(D)$\pi \int_{0}^{4} \left( 3 - \sqrt{y} \right)^2 \, dy$

10. Let $R$ be the region enclosed by the graph of $y = 4 - x^2$ and the $x$-axis. Which of the following definite integrals gives the volume of the solid generated when region $R$ is revolved about the horizontal line $y = -3$?

(A)$\pi \int_{-2}^{2} \left( (7 - x^2)^2 - 9 \right) \, dx$(B)$\pi \int_{-2}^{2} \left( (1 - x^2)^2 - 9 \right) \, dx$(C)$\pi \int_{-2}^{2} \left( (4 - x^2)^2 - 9 \right) \, dx$(D)$\pi \int_{-2}^{2} \left( (7 - x^2)^2 - (4 - x^2)^2 \right) \, dx$

11. A particle moves along the x-axis with velocity given by $v(t) = 3t^2 - 12$ for time $t \ge 0$. What is the total distance traveled by the particle from time $t = 0$ to time $t = 3$?

(A)9(B)16(C)23(D)25

12. A particle moves along the x-axis with velocity given by $v(t) = t^2 - 6t + 8$ for $0 \le t \le 6$. Which of the following expressions gives the total distance traveled by the particle from $t = 0$ to $t = 5$?

(A)$\int_0^5 (t^2 - 6t + 8) dt$(B)$\int_0^2 (t^2 - 6t + 8) dt - \int_2^4 (t^2 - 6t + 8) dt + \int_4^5 (t^2 - 6t + 8) dt$(C)$\int_0^2 (t^2 - 6t + 8) dt + \int_2^4 (t^2 - 6t + 8) dt + \int_4^5 (t^2 - 6t + 8) dt$(D)$-\int_0^2 (t^2 - 6t + 8) dt + \int_2^4 (t^2 - 6t + 8) dt - \int_4^5 (t^2 - 6t + 8) dt$

13. The rate at which rain accumulates in a gauge is modeled by the function $R$, where $R(t)$ is measured in centimeters per hour and $t$ is measured in hours since midnight. Which of the following is the best interpretation of the statement $\int_2^5 R(t) \, dt = 4.5$ ?

(A)The total amount of rain that accumulated in the gauge between 2:00 a.m. and 5:00 a.m. is 4.5 centimeters.(B)The amount of rain in the gauge increased by 4.5 centimeters per hour between 2:00 a.m. and 5:00 a.m.(C)The rate at which rain is accumulating in the gauge is 4.5 centimeters per hour at time $t=5$.(D)The average rate of rainfall between 2:00 a.m. and 5:00 a.m. was 1.5 centimeters per hour.

14. A particle moves along the $x$-axis with velocity given by $v(t) = 4t^3 - 6t^2$ for time $t \geq 0$. If the position of the particle is $x = 7$ at time $t = 1$, what is the position of the particle at time $t = 2$ ?

(A)1(B)6(C)8(D)9

15. Let $R$ be the region enclosed by the graphs of $f(x) = -x^2 + 2x + 3$ and $g(x) = x^2 - 4x + 3$. Which of the following is the area of $R$?

(A)$\frac{9}{2}$(B)9(C)18(D)$\frac{27}{2}$

16. Which of the following definite integrals represents the area of the region enclosed by the graphs of $y = \sin(x)$ and $y = \cos(x)$ between the lines $x = 0$ and $x = \pi$?

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

17. Let $R$ be the region in the $xy$-plane bounded by the graphs of $x = y^2 - 3$ and $x = 2y$. Which of the following definite integrals represents the area of region $R$?

(A)$\displaystyle \int_{-1}^{3} $(y^2 - 2y - 3) $\, $dy(B)$\displaystyle \int_{-1}^{3} $(2y - y^2 + 3) $\, $dy(C)$\displaystyle \int_{-3}^{6} $(2y - y^2 + 3) $\, $dy(D)$\displaystyle \int_{-3}^{6} $(y^2 - 2y - 3) $\, $dy

18. Let $f$ and $g$ be continuous functions such that $f(x) \ge g(x)$ on the interval $[a, b]$ and $g(x) \ge f(x)$ on the interval $[b, c]$. Which of the following gives the total area of the region bounded by the graphs of $f$ and $g$ for $a \le x \le c$?

(A)$\displaystyle \int_a^c (f(x) - g(x)) \, dx$(B)$\displaystyle \int_a^c |f(x) + g(x)| \, dx$(C)$\displaystyle \int_a^b (f(x) - g(x)) \, dx + \int_b^c (g(x) - f(x)) \, dx$(D)$\displaystyle \int_a^b (g(x) - f(x)) \, dx + \int_b^c (f(x) - g(x)) \, dx$

19. Which of the following expressions gives the area of the region bounded by the graphs of $y = x^3$ and $y = 4x$?

(A)$\displaystyle \int_{-2}^2 (4x - x^3) \, dx$(B)$\displaystyle \int_{-2}^0 (4x - x^3) \, dx + \int_0^2 (x^3 - 4x) \, dx$(C)$\displaystyle \int_{-2}^0 (x^3 - 4x) \, dx + \int_0^2 (4x - x^3) \, dx$(D)$\displaystyle \int_{0}^2 (4x - x^3) \, dx - \int_{-2}^0 (4x - x^3) \, dx$

20. Let $R$ be the region in the first quadrant bounded by the graphs of $y = 2x$ and $y = x^2$. Which of the following definite integrals represents the volume of the solid whose base is region $R$ and whose cross sections perpendicular to the $x$-axis are squares?

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

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

(A)8(B)$8\pi$(C)$\frac{16}{3}$(D)$\frac{64}{3}$

22. Let $R$ be the region in the first quadrant bounded by the graph of $y = 4 - x^2$ and the $x$-axis. 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 the $xy$-plane. Which of the following integrals gives the volume of the solid?

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

23. The base of a solid is the region bounded by the line $x = 1$, the line $x = 3$, the graph of $y = e^x$, and the $x$-axis. For this solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in the $xy$-plane. What is the volume of the solid?

(A)$\frac{1}{4}(e^6 - e^2)$(B)$\frac{1}{2}(e^6 - e^2)$(C)$\frac{1}{2}(e^3 - e)^2$(D)$\frac{1}{4}(e^9 - e^3)$

24. Let $R$ be the region in the first quadrant bounded by the graph of $y = \ln(x)$, the $y$-axis, the horizontal line $y=1$, and the horizontal line $y=3$. Which of the following definite integrals gives the volume of the solid generated when $R$ is revolved about the $y$-axis?

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

25. A particle moves along the x-axis with velocity given by $v(t) = 3t^2 - 12t + 9$ for time $t \geq 0$. If the particle is at position $x = 5$ at time $t = 0$, what is the total distance traveled by the particle from $t = 0$ to $t = 2$?

(A)2(B)6(C)7(D)11

26. Let $R$ be the region in the first quadrant bounded by the graphs of $y = \sqrt{x}$ and $y = \frac{x}{2}$. Which of the following definite integrals represents the volume of the solid generated when region $R$ is revolved about the vertical line $x = 6$ ?

(A)$\pi \int_{0}^{2} \left[ (6 - y^2)^2 - (6 - 2y)^2 \right] \, dy$(B)$\pi \int_{0}^{2} \left[ (6 - 2y)^2 - (6 - y^2)^2 \right] \, dy$(C)$\pi \int_{0}^{4} \left[ (6 - \frac{x}{2})^2 - (6 - \sqrt{x})^2 \right] \, dx$(D)$\pi \int_{0}^{2} \left[ (y^2 + 6)^2 - (2y + 6)^2 \right] \, dy$

27. A particle moves along the $x$-axis with velocity given by $v(t) = 3t^2 - 12t$ for time $t \ge 0$. What is the total distance traveled by the particle from time $t = 0$ to time $t = 5$?

(A)25(B)32(C)39(D)57

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

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

B. Short Answer Questions (2 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 29:

Question 30:

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