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prac3bsol

# prac3bsol - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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± ± ± �� 18.02 Practice Exam 3 B Solutions (1,2) y = 2 x x = 1 ± ± (1,1) b) 1 y dxdy + 2 1 dxdy. 1. a) ± 0 y/ 2 1 y/ 2 ±� y = x (the first integral corresponds to the bottom half 0 y 1, the second ± integral to the top half 1 y 2.) r sin 2. a) �dA = rdrd� = sin �drd� . 2 r �� 3 M = �dA = sin drd� = 2 sin �d� = 2 cos = 4 . 0 R 0 1 0 1 �� 1 3 b) ¯ x = x�dA = r cos sin �drd� M R 4 0 1 The reason why one knows that ¯ x = 0 without computation is that the region and the density are symmetric with respect to the y -axis ( ( x, y ) = ( x, y )). 3. a) N x = 12 y = M y , hence F is conservative. b) f x = 3 x 2 6 y 2 f = x 3 6 y 2 x + c ( y ) f y = 12 xy + c ( y ) = 12 xy + 4 y . So c ( y ) = 4 y , thus c ( y ) = 2 y 2 (+ constant). In conclusion f =
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