quiz3-T-sol

quiz3-T-sol - Quiz 3 - Tuesday (math 32b) November 14, 2007...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Page1 / 2

quiz3-T-sol - Quiz 3 - Tuesday (math 32b) November 14, 2007...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online