080205quiz4solution

# 080205quiz4solution - -x 2 = x 2-12 gives us x =-4 , +4....

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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 | ² = -∞ 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 - x 2 and y = x 2 - 12. We ﬁnd at ﬁrst the points of intersection. Setting 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 = Z 2 [(2 x ) 2-( x 2 ) 2 ] dx = Z 2 4 x 2-x 4 dx = ( 4 3 x 3-1 5 x 5 ) | 2 = 64 15...
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## 080205quiz4solution - -x 2 = x 2-12 gives us x =-4 , +4....

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