Same Idea as last time. Need you online at anytime with me to finish the assignment on March 4th. The assignment is timed and I will need work and answers. This time is around 9 questions.
Math 141 Exam 2 Review Problems (Answers at end of document) 1. Find the area of the region bounded by y= ex , y = 12 ex and the y-axis. Include a sketch of the region and label the points of intersection. 2. Find the area of the region bounded by x= 8 y2 and x= y2 . Include a sketch of the region and label the points of intersection. 3. Find the area of the region bounded by y= p x ,y = x= 2 and the line x= 5. Include a sketch of the region and label the points of intersection. 4. Find the area of the region bounded by y= sin x, y = cos xand the lines x= 0 and x = = 3. Include a sketch of the region and label the points of intersection. 5. Find the average value of f(x ) = xcos xon the interval [0 ;2 ]. 6. Find the average value of f(x ) = xe3 x on the interval [0 ;4]. 7. The temperature (in degrees Celsius) at time t(in hours) in an art museum varies according to T(t) = 20 + 5 cos( 12 t ). Find the average temperature over the time period [2 ;6]. 8. Find the average speed over the time interval [1 ;5] (time in seconds) of a particle whose position at time tis s(t) = t3 6t2 . 9. Consider a pyramid with square base of side length 10 cm and height 15 cm. (a) Slice the pyramid into horizontal slices each with height xcm. Find the ap- proximate volume of the slice that is a vertical distance x i cm from the tip of the pyramid. The only variables in your answer should be x i and x. (b) Write a de nite integral that gives the volume of the pyramid. DO NOT EVAL- UATE THE INEGRAL. 10. Consider a sphere with radius 3 m. (a) Slice the sphere into horizontal slices each with height ym. Find the approx- imate volume of the slice that is y i m above the bottom of the sphere. The only variables in your answer should be y i and y. (b) The sphere is a tank that has water to a height of 2 m above the bottom of the sphere. Write a de nite integral that gives the volume of the water in the tank. DO NOT EVALUATE THE INEGRAL. 11. Find the volume of a solid whose base is the semicircle y= p 9 x2 , where 3 x 3, and whose cross sections perpendicular to the x-axis are squares. 12. Find the volume of a solid whose base is the region enclosed by y= x2 and y= 3, and whose cross sections perpendicular to the y-axis are squares. 13. Consider the region in the rst quadrant bounded by y= x4 , y = 16 and the y-axis. (a) Sketch the region and label the points of intersection. (b) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the line y= 20. (c) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the x-axis. 14. Consider the region in the rst quadrant bounded by x= y3 , x = 8, and the x-axis. (a) Sketch the region and label the points of intersection. (b) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the line x= 1. (c) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the y-axis. 15. Consider the region in the rst quadrant bounded by y= x4 , y = 16 and the y-axis. (a) Sketch the region and label the points of intersection. Page 2 (b) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the y-axis. (c) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the x-axis. (d) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the line x= 2. 16. Consider the region in the rst quadrant bounded by x= y2 , x = 4, and the x-axis. (a) Sketch the region and label the points of intersection. (b) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the y-axis. (c) Set up, but do not evaluate, a de nite integral to nd the volume of the solid obtained by rotating the region about the x-axis. (d) Set up, but do not evaluate, a de nite integral to nd the volume of the ob- tained by rotating the region about the line y= 5. 17. Evaluate lim x! 0x sin x 2 x 3 . 18. Evaluate lim x!1 e x x 2. 19. Evaluate lim x! 0+ p x ln x. 20. (a) Evaluate lim x!1 [ln(2 x+ 3) ln(5 x2 + 7)]. (b) Evaluate lim x!1 [ln(2 x+ 3) ln(5 x+ 7)]. 21. (a) Evaluate lim x! 0+ x x . (b) Evaluate lim x! 0+ x ln x . 22. Evaluate lim x! 0+ ( e 2 x + x)1 =x . Page 3 23. (a) Evaluate lim x!1 x 1 =x 2 . (b) Evaluate lim x!1 xln 3 1+ln x . 24. (a) Evaluate Z 1 3 1 x 2 + 9 dx or show that it diverges. (b) Evaluate Z 1 1 1 x 2 + 9 dx or show that it diverges. 25. Evaluate Z 6 5 1 ( x + 1) 2dx or show that it diverges. 26. Evaluate Z 5 0 p x ln x dx or show that it diverges. 27. Evaluate Z 1 0 x 2 e x3 dx or show that it diverges. Page 4 Answers 1. 12 ln 6 10 2. 64 3 3. 107 12 2 3 (5) 3 =2 4. 2 p 2 3 2 p 3 2 5. 0 6. 1 36 11 e12 + 1 7. 20 + 15 2 8. 17 2 m/s 9. (a) 4 9 ( x i) 2 x (b) Z 15 0 4 9 x 2 dx 10. (a) [9 (3 y i) 2 ] y (b) Z 2 0 [9 (3 y)2 ] dy 11. 36 12. 18 13. (a) Graph (b)Z 2 0 (20 x4 )2 16 dx (c) Z 2 0 256 x8 dx 14. (a) Graph (b)Z 2 0 [81 (y 3 + 1) 2 ] dy (c) Z 2 0 (64 y6 ) dy Page 5 15. (a) Graph (b)Z 2 0 2 x (16 x4 ) dx (c) Z 2 0 (256 x8 ) dx (d) Z 2 0 2 (x + 2)(16 x4 ) dx 16. (a) Graph (b)Z 4 0 2 (x p x ) dx (c) Z 4 0 (p x )2 dx (d) Z 4 0 [5 2 (5 p x )2 ] dx 17. 1 12 18. 1 19. 0 20. (a) 1 (b) 2 5 21. (a) 1 (b)e 22. e3 23. (a) 1 (b) 3 24. (a) 12 (b) 3 25. Diverges 26. 2 3 5 3 =2 ln 5 4 9 5 3 =2 27. 1 3 Page 6