Name Review for Test: Applications of Integration AP Calculus AB Test Topics: Mean Value Theorem for Integrals (section 4.4) Average Value of a Function (manipulation of MVT for Integrals) (section 4.4) Net change (including displacement, total distance, and current position) (section 4.4) Integration by Substitution (section 4.5) Differential Equations and Slope Fields (section 6.1) Exponential Growth & Decay (section 6.2) Area between two curves (section 7.1) Volume (disk method, washer method, and cross sections) (sections 7.2 & 7.3) Review Questions (in no particular order ) 1. Find the volume of a solid formed by revolving the region bounded by y x, y = 0, x = 0, and x = 2 about the x axis. 2. Find the volume of revolving the above shape (#1) about the y axis. 3. Rotate the region in the first quadrant bounded by y x, y = 1, and x = 0 about the y axis. Find the volume. 4. Rotate the above region (#3) about the line y = 5, and find the volume.
5. Find the volume of the solid formed by revolving the region bounded by y x, and y x about the x axis. 6. Find the volume of the solid whose base is the region bounded by the lines y x4, x = 4, x = 0, and y = 0, if the cross sections taken perpendicular to the x axis are semicircles. 7. Find the volume of the solid whose base is bounded by y2x and y and the cross section perpendicular to the y axis are squares. Find its volume. 8. The base of a solid is the region enclosed by the semicircle y 25 x 2 and the x axis. Cross sections perpendicular to the x axis are squares. Find the volume of this region.
9. Find the area enclosed by y x and y x2 on 0, 2. 10. Find the area enclosed by fx x, gx 2x5, and the x axis. 11. The acceleration of a particle is given by at 4t. If the initial velocity is 1 m/s, find the displacement of the particle in the first 4 seconds and the total distance traveled by the particle in the first 4 seconds. 12. Below is a graph showing the velocity of a particle in m/s. It starts with an initial position of 50. Based on the graph: a. What is the particle s displacement in 18 seconds? b. What is the total distance traveled by the particle in 18 seconds? c. Give the particle s position at time t 18.
13. Find the average value of the function fx 4xx over 0, 3. 14. The velocity function for a particle moving in a straight line is vt t 7t10. (a) Find the displacement traveled by the particle for the time interval 1, 7. (b) Find the total distance traveled by the particle for the time interval 1, 7. 15. Find the exact value of xe dx
16. Evaluate the indefinite integral: xe 1 2e 4x dx 17. The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. a. Find the initial population. b. Write an exponential growth model for the bacteria population. Let t represent the time in hours. c. Use the model to determine the number of bacteria after 8 hours. d. After how many hours will the bacteria count be 25,000?
Answers: 1. V πx dx π cubic units 2. V πy dy 8π cubic units 3. V πy dy cubic units 4. V π 5 x 5. V πx x dx 5 1 dx 3.429π cubic units π cubic units 6. V dx π 18.667π cubic units 7. V 2y dy 9.752 cubic units 8. V 25 x dx 9. A x dx. 10. A x dx 11. st vtdt 2 x dx 12. a) 84 60 144 b) 204 60 144.. c) 134 50 60 144 13. 3 4x x dx 14. a) 6 b) 15 15. 16. ln2e 4xC 17. a) 45 bacteria b) y 45e c) 2,766 bacteria d) 12.276 hours cubic units sq units 2x 5dx 3.370 sq units 2t 1 dx 46.667 m