Review for Mathematics Test: Integrals and Series - Prof. David Glickenstein, Exams of Calculus

Download Review for Mathematics Test: Integrals and Series - Prof. David Glickenstein and more Exams Calculus in PDF only on Docsity! Review for Test 2 1. Chapter 7 Check Your Understanding: 18, 20, 25,27 18. True 20. False. Try f (x) = 1x+1 : 25. True. Substitute w = ax: 27. False. 2. Chapter 8 Check Your Understanding: 1, 2, 3, 4,13, 15 1. True 2. False 3. False 4. True 13. False 15. False 3. Chapter 9 Check Your Understanding: 14, 22, 23, 24, 25 14. False. The terms do not go to zero. 22. True. 23. False. Consider the harmonic series. 24. False. Consider an = bn = 1n : 25. False. Same example as 24. 4. Chapter 9 Review: 16 16. a. 0:23232323 : : : b. the sum is 2399 (use geometric series formula). 5. Chapter 8 Review: 7, 11, 13, 19, 21 These answers are all in the back of the book. 6. Show that the following integrals converge or diverge (you must actually show the comparison, not just give an idea of why you think it is based on looking at highest powers, etc): a) R1 2 1 x+1+sin xdx 1 x+ 1 + sinx  1 x+ 2 hence Z 1 2 1 x+ 1 + sinx dx  Z 1 2 1 x+ 2 dx which diverges, hence this integral diverges. b) R1 1 1 x10+2xdx on [1;1) we see that 1 x10 + 2x  1 x10 1 and since R1 1 1 x10 dx converge, this integral must converge. c) R 2 0 1 x3+2dx This integral is only improper at the cube root of 2; which is not in the interval. Hence this integral converges. d) R1 1 ex 2 dx On this interval x2  x x2  x ex 2  ex and since R1 1 exdx converges, this integral converges. 7. Write an integral that represents the arclength of the following curves: a) y = sinx between x = 0 and x = 2:Z 2 0 p 1 + cos2 xdx b) y = x3 + 1 between x = 1 and x = 1:Z 1 1 p 1 + 9x4dx c) the parametric curve x = e2t, y = cos t for t 2 [0; 3] :Z 3 0 p 4e4t + sin2 tdt d) the ellipse x = 2 cos t; y = 4 sin t for t 2 [0; 2] :Z 2 0 p 4 sin2 t+ 16 cos2 tdt e) y = lnx between x = 1 and x = 2:Z 2 1 r 1 + 1 x2 dx 8. Let R be the region bounded by the curve y = x2, the x-axis, and the lines x = 1 and x = 2: Compute the volumes of the following solids: a) the solid de…ned by rotating R about the y-axis.  Z 4 0 (16 y) dy = 56 2