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# A03S - Math 138 Assignment 3 Solutions 1 The plot of x = ey...

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Math 138 Assignment 3 Solutions 1. The plot of x = e y - y 4 - 1 is shown below. -2 -1 0 1 2 -1 1 b Figure 1: x = e y - y 4 - 1 To rotate this about the x -axis we must use cylindrical shells as we can’t solve for y in terms of x . Note that b is a root of e y - y 4 - 1. The typical rectangle to create each shell is shown below -2 -1 0 1 2 -1 1 b dy y

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The volume is thus V = Z b 0 2 π y |{z} radius x |{z} height dy = 2 π Z b 0 y ( e y - y 4 - 1) dy = 2 π ye y - e y - y 6 6 - y 2 2 ! b 0 using integration by parts = 2 π e b ( b - 1) - b 6 6 - b 2 2 + 1 ! 2. (a) Note the integrand is undefined at x = 0 so we split it up into two Z 1 - 1 e x e x - 1 dx = Z 0 - 1 e x e x - 1 + Z 1 0 e x e x - 1 = lim b 0 - Z b - 1 e x e x - 1 dx + lim c 0 + Z 1 c e x e x - 1 dx = lim b 0 - ln | e x - 1 | b - 1 + lim c 0 + ln | e x - 1 | 1 c Because lim b 0 - ln | e b - 1 | → -∞ , the first integral diverges and thus the original integral diverges (we needn’t look at the other values). (b) Before rewriting with limits we first convert the integral by letting u = x then du = 1 2 x dx and the integral becomes Z 0 1 x (1 + x ) dx = 2 Z 0
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A03S - Math 138 Assignment 3 Solutions 1 The plot of x = ey...

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