53 Worksheet Solutions 10_21

53 Worksheet Solutions 10_21 - Math 53: Multivariable...

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Math 53: Multivariable Calculus Solutions for Worksheet 10/21/11: Double Integrals? Pshaw! TRIPLE! Exercise 0.1. Evaluate the triple integral Z 1 0 Z z 0 Z y 0 ze - y 2 dxdydz Solution. Z 1 0 Z z 0 Z y 0 ze - y 2 dxdydz = Z 1 0 Z z 0 xze - y 2 ± ± ± y x =0 dydz = Z 1 0 Z z 0 yze - y 2 dydz = Z 1 0 - 1 2 ze - y 2 ± ± ± ± z 0 dydz = - 1 2 Z 1 0 ² ze - z 2 - ze - 0 2 ³ dydz = - 1 2 Z 1 0 ² ze - z 2 - z ³ dydz = - 1 2 ´ - 1 2 e - z 2 - z 2 2 µ± ± ± ± 1 0 = 1 4 ² e - z 2 + z 2 ³ ± ± ± ± 1 0 = 1 4 ( e - 1 + 1 - e 0 - 0 ) = 1 4 ( e - 1 ) = 1 4 e . ± Exercise 0.2. Find RRR T xdV , where T is bounded by the parabloid x = 4 y 2 + 4 z 2 and the plane x = 4. 1
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2 Solution. We can write T as 4 y 2 + 4 z 2 x 4, - 1 - z 2 y 1 - z 2 , and - 1 z 1. Our integral is then Z 1 - 1 Z 1 - z 2 - 1 - z 2 Z 4 4 y 2 +4 z 2 xdxdydz = Z 1 - 1 Z 1 - z 2 - 1 - z 2 x 2 2 ± ± ± ± 4 4 y 2 +4 z 2 dydz = Z 1 - 1 Z 1 - z 2 - 1 - z 2 ² 16 2 - (4 y 2 + 4 z 2 ) 2 2 ³ dydz =8 Z 1 - 1 Z 1 - z 2 - 1 - z 2 ( 1 - ( y 2 + z 2 ) 2 ) dydz. To save ourselves from a whole lot of messy integrating-high-powers-
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53 Worksheet Solutions 10_21 - Math 53: Multivariable...

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