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

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39. 40. 41–46 |||| Sketch the region and find its area (if the area is finite). 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 infinite and the integrand has an infinite 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 ² dx ± y ² 1 1 s x ± 1 ± x ² y ² 0 1 s x ± 1 ± x ² ³ 0, ² ² y ² 0 1 s x ± 1 ± x ² y 1 0 e ³ x s x y ´ /2 0 dx x sin x y ² 1 x s 1 ± x 6 y ² 1 dx x ± e 2 x 51. y ² 1 2 ± e ³ x x y ² 1 cos 2 x 1 ± x 2 x ² 2 t ± x ² 2 µ x µ 20 t f x ² 2 t ± x ² f ± x ² ± 1 ´ s x x ² 2 t ± x ² t ± 5 x t 2 t ± x ² t ± x ² ± 1 ´ ( s x ³ 1) x ² 1 t ± x ² 1 µ x µ 10 t f x ² 1 t ± x ² f ± x ² ± 1 ´ x 2 x ² 1 t ± x ² t ± 2 x t 1 t ± x ² t ± x ² ± ± sin 2 x ²´ x 2 S ± { ± x , y ² µ ³ 2 x µ 0, 0 µ y µ 1 ´ s x ± 2 } S ± ¶± x , y ² µ 0 µ x ´ 2, 0 µ y µ sec 2 x · S ± ¶± x , y ² µ x · 0, 0 µ y µ x ´± x 2 ± 9 ²· S ± ¶± x , y ² µ 0 µ y µ 2 ´± x 2 ± 9 ²· 43.
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Calculus 5e_Part270 - 538 CHAPTER 7 TECHNIQUES OF...

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