# Changing the variable in multiple integrals

by kidsmoker
Tags: integrals, multiple, variable
 P: 88 1. The problem statement, all variables and given/known data Evaluate $$\int\int(x-y)^2sin^2(x+y)dxdy$$ taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi). 2. Relevant equations $$I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv$$ where J is the Jacobian. 3. The attempt at a solution Okay so I've just been learning this for the first time, so I may be doing it completely wrong! I used the transformations u=x-y, v=x+y which give the Jacobian as 2. Now i wasn't sure how to get the new limits for the integrals. What I did was apply the transformation above to give new vertices: (pi,0) -> (pi,pi) (0,pi) -> (-pi,pi) (pi,2pi) -> (-pi,3pi) (2pi,pi) -> (pi,3pi) This gives a simple rectangle, so then i just wrote $$I = 2*\int^{3\pi}_{\pi}\int^{\pi}_{-\pi}u^2sin^2(v)dudv = \frac{4\pi^{4}}{3}.$$ I wish this was right, but i've a feeling it's not :-( Any help greatly appreciated!
 Sci Advisor HW Helper Thanks P: 25,243 The rectangle looks ok. But haven't you got the jacobian factor upside down?
 P: 88 Ah yeah, should be 1/2. Other than that though does my method look correct? Thanks.