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Unformatted text preview: Quiz 3 - Tuesday (math 32b) November 14, 2007
f (x, y)dxdy in Problem 1. For a function f (x, y), rewrite the integral S variables (r, ). Here the transformation (r, ) (x, y) is given by x = r cos3 , y = r sin3 and S is the area bounded by the curve x2/3 + y 2/3 = 1. Solution. First, find the image of the region under the given transformation: r2/3 cos2 + r2/3 sin2 = r
2/3 1 1 1 r = = Thus, image is the disc of radius 1 centered at the origin. Second, compute the Jacobian: (x, y) (u, v) = = = det cos3 sin3 2 2 -3r cos2 sin 3r sin2 cos 2 2 = = = 3r sin2 cos2 . (3r(sin 2 cos4 + sin4 cos2 )) 3r sin cos (cos + sin ) Thus, the integral is equivalent to
2 1 I=
0 0 f (r cos3 , r sin3 ) 3r sin2 cos2 drd. 1 Problem 2. Find the integral C xdy - ydx, where C is the part of the parabola y = x2 between the points (0, 0) and (2, 4). Solution. Parametric equation of the curve is x = t, y = t2 , 0 t 2. The integral is 2 2 8 t2 dt = . I= t 2tdt - t2 dt = 3 0 0 0 2 ...
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This test prep was uploaded on 04/14/2008 for the course MATH 32B taught by Professor Rogawski during the Fall '08 term at UCLA.
- Fall '08