sol1 - Math 264: Advanced Calculus Winter 2008 Assignment 1...

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Unformatted text preview: Math 264: Advanced Calculus Winter 2008 Assignment 1 due Thursday, January 17 Every problem is worth 5 points. Due to time constraints, some problems may not be marked. Problem 1 (Adams, 14.2 # 18). Sketch the domain of integration and compute the iterated integral: R 1 dx R x 1 / 3 x p 1- y 4 dy . Solution: We change the order of integration, to find that the integral (which we denote by I ) can be rewritten as I = Z 1 y =0 p 1- y 4 dy Z y x = y 3 dx = Z 1 y p 1- y 4 dy- Z 1 y 3 p 1- y 4 dy. Call the first integral I 1 , and the second integral I 2 . I 2 can be evaluated as follows: since [(1- y 4 ) 3 / 2 ] =- 6 y 3 (1- y 4 ) 1 / 2 , we have I 1 = (1 / 6) R 1 d [(1- y 4 ) 3 / 2 ] = (1 / 6)(1- y 4 ) 3 / 2 | 1 y =0 =- 1 / 6 . To evaluate I 1 , we change variables u = y 2 , and write I 1 = R 1 y p 1- y 4 dy = (1 / 2) R 1 u =0 1- u 2 du = 1 2 [ u 2 1- u 2 + 1 2 arcsin u ] 1 u =0 = / 8. The answer is thus I = / 8- 1 / 6. Problem 2 (Adams, 14.3 # 11). Compute R R Q e- xy dA , where Q is the first quadrant in the xy-plane....
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sol1 - Math 264: Advanced Calculus Winter 2008 Assignment 1...

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