Unformatted text preview: olving about the y-axis. 49. Let be the region bounded by the curve y = x−1/4 and the x-axis, 0 < x ≤ 1. (a) Sketch . (b) Find the area of . (c) Find the volume of the solid obtained by revolving about the x-axis. (d) Find the volume of the solid obtained by revolving about the y-axis. 50. Prove the validity of the comparison test (10.7.2). In Exercises 51–56, use the comparison test (10.7.2) to determine whether the integral converges.
2 diverges. Thus, the improper integral 2x dx 1 + x2 −∞ diverges. (b) Show that lim 58. Show that
∞ b→∞ ∞ 2x dx = 0. 2 −b 1 + x and b (a)
−∞ sin x dx diverges
b (b) lim b→∞ −b sin x dx = 0. 59. Calculate the arc distance from the origin to the point (x(θ1 ), y(θ1 )) along the exponential spiral r = a ecθ . (Take a > 0, c > 0.) 60. The function 1 f...
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