Math 2E 44275 Spring 2014 Quiz 2 4 17 14 Name Print ID Number Show all of your work Partial credit will be given 1 10 points Evaluate ZZ x2 dA R where R is the region bounded by the ellipse 9x2 4y2 36 Solution 1 Letting x 2u y 3v ZZ R ZZ cid 20 1 u2 v2 1 x2 4u2 and the planar ellipse 9x2 4y2 36 is the image of the disk u2 v2 1 Thus x2dA 4u2 6 dudv 24r2 cos2 rdrd 24 cos2 d r3dr Z 2 0 Z 1 0 24 2 x 1 4 sin 2x 24 6 1 4 Solution 2 Since 9x2 4y2 36 the standard equation of the ellipse is cid 12 cid 12 cid 12 cid 12 6 x y u v 0 3 cid 12 cid 12 cid 12 cid 12 2 0 Z 2 Z 1 cid 21 1 cid 20 1 0 0 4 r4 0 cid 21 2 0 x2 4 y2 9 1 cid 12 cid 12 cid 12 cid 12 2 cos 2r sin 3r cos 3 sin Let x 2r cos y 3r sin where 0 r 1 0 2 Therefore Thus ZZ x2dA R x y r Z 2 Z 1 cid 20 1 0 0 2r cos 2 6r drd 24 cos2 d r3dr 24 2 x 1 4 sin 2x 24 6 1 4 cid 21 2 cid 20 1 4 r4 0 cid 21 1 0 cid 12 cid 12 cid 12 cid 12 6r Z 2 0 Z 1 0
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