quiz09' - v = 1, v = 8, v = u 3 , and v = 8 u . Also, ( x,y...

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Quiz 9 Solutions 1. Evaluate the integral R 1 0 R 1 - y 2 0 R x 2 + y 2 x 2 + y 2 xyz dz dx dy . [Hint: change to cylindrical coordinates.] The region given consists of the part of the circle in the first quadrant in the xy -plane, and goes from r 2 to r in the z direction. So, our integral is Z 1 0 Z π 2 0 Z r r 2 r 3 sin θ cos θz dz dθ dr = 1 4 Z 1 0 Z π 2 0 r 3 sin(2 θ )( r 2 - r 4 ) dθ dr = 1 4 Z 1 0 r 5 - r 7 dr Z π 2 0 sin(2 θ ) = 1 4 ± 1 6 - 1 8 ² = 1 96 . 2. Find the area of the region bounded by xy = 1, xy = 8, y = x 2 , and y = 8. Let x = u and y = v u . Then our bounding curves become
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Unformatted text preview: v = 1, v = 8, v = u 3 , and v = 8 u . Also, ( x,y ) ( u,v ) = 1 u . So, the area of the region is Z 8 1 Z 3 v v 8 1 u du dv = Z 8 1 ln 3 v-ln( v/ 8) dv = Z 8 1 ln 8-2 3 ln v dv = v ln 8-2 3 ( v ln v-v ) 8 v =1 = 8 ln 8-16 3 ln 8 + 16 3-ln 8-2 3 = 5 ln 2 + 14 3 . 1...
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