Calculus 5e_Part270

# Calculus 5e_Part270 - 538 CHAPTER 7 TECHNIQUES OF...

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39. 40. 41–46 |||| Sketch the region and fi nd its area (if the area is fi nite). 41. 42. ; ; 44. ; 45. ; 46. ; 47. (a) If , use your calculator or computer to make a table of approximate values of for , 5, 10, 100, 1000, and 10,000. Does it appear that is convergent? (b) Use the Comparison Theorem with to show that is convergent. (c) Illustrate part (b) by graphing and on the same screen for . Use your graph to explain intuitively why is convergent. ; 48. (a) If , use your calculator or computer to make a table of approximate values of for , 10, 100, 1000, and 10,000. Does it appear that is convergent or divergent? (b) Use the Comparison Theorem with to show that is divergent. (c) Illustrate part (b) by graphing and on the same screen for . Use your graph to explain intuitively why is divergent. 49–54 |||| Use the Comparison Theorem to determine whether the integral is convergent or divergent. 49. 50. 52. 53. 54. 55. The integral is improper for two reasons: The interval is in fi nite and the integrand has an in fi nite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: y 1 0 1 s x 1 x dx y 1 1 s x 1 x dx y 0 1 s x 1 x dx 0, y 0 1 s x 1 x dx y 1 0 e x s x dx y /2 0 dx x sin x y 1 x s 1 x 6 dx y 1 dx x e 2 x 51. y 1 2 e x x dx y 1 cos 2 x 1 x 2 dx x 2 t x dx 2 x 20 t f x 2 t x dx f x 1 s x x 2 t x dx t 5 x t 2 t x dx t x 1 ( s x 1 ) x 1 t x dx 1 x 10 t f x 1 t x dx f x 1 x 2 x 1 t x dx t 2 x t 1 t x dx t x sin 2 x x 2 S { x , y 2 x 0, 0 y 1 s x 2 } S x , y 0 x 2, 0 y sec 2 x S x , y x 0, 0 y x x 2 9 S x , y 0 y 2 x 2 9 43.

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