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080205quiz4solution

# 080205quiz4solution - -x 2 = x 2-12 gives us x =-4 4 Thus A...

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MATH 126 QUIZ #4 02/21/08 Mathias Knape Last Name : Circle discussion section First Name: 11 am 12 pm 1 pm Note: In order to receive full credit, please show all your work (conceptual steps, definitions you use, ...) 1. Determine if the integral is convergent or divergent. If convergent, compute the value of the integral. (a) Z 1 - 1 e x e x - 1 dx Notice that f ( x ) = e x e x - 1 has an infinite discontinuity at x = 0. Thus we have to split up the integral in two parts. Z 1 - 1 e x e x - 1 dx = Z 0 - 1 e x e x - 1 dx + Z 1 0 e x e x - 1 dx Let us consider Z 0 - 1 e x e x - 1 dx . Z 0 - 1 e x e x - 1 dx = lim t 0 - Z t - 1 e x e x - 1 dx = lim t 0 - [ln | e x - 1 | ] t - 1 = lim t 0 - ln | e t - 1 | - ln | e - 1 - 1 | = -∞ Thus Z 0 - 1 e x e x - 1 dx is divergent and hence Z 1 - 1 e x e x - 1 dx is divergent. 2. Find the area of the region bounded by y = 20

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Unformatted text preview: -x 2 = x 2-12 gives us x =-4 , +4. Thus A = Z 4-4 [(20-x 2 )-( x 2-12)] dx = Z 4-4 32-2 x 2 dx = 2 Z 4 32-2 x 2 dx = 2[32 x-2 3 x 3 ] 4 = 2(128-128 3 ) = 512 3 . 3. (a) Consider the region enclosed by y = 2 x and y = x 2 . Sketch the region and the solid obtained by rotating the region about the x-axis and a typical disk or washer. (b) Find the volume of the solid. V = π Z 2 [(2 x ) 2-( x 2 ) 2 ] dx = π Z 2 4 x 2-x 4 dx = π ( 4 3 x 3-1 5 x 5 ) | 2 = 64 π 15...
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080205quiz4solution - -x 2 = x 2-12 gives us x =-4 4 Thus A...

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