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

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Unformatted text preview: Math 138 Assignment 3 Solutions 1. The plot of x = e y- y 4- 1 is shown below.-2-1 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 cant 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 1 2-1 1 b dy y The volume is thus V = Z b 2 y |{z} radius x |{z} height dy = 2 Z b y ( e y- y 4- 1) dy = 2 ye y- e y- y 6 6- y 2 2 ! b using integration by parts = 2 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- 1 e x e x- 1 + Z 1 e x e x- 1 = lim b - Z b- 1 e x e x- 1 dx + lim c + Z 1 c e x e x- 1 dx = lim b - ln | e x- 1 | b- 1 + lim c + ln | e x- 1 | 1 c Because lim b - ln | e b- 1 | - , the first integral diverges and thus the original integral diverges (we neednt look at the other values)....
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A03S - Math 138 Assignment 3 Solutions 1. The plot of x = e...

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