# Changing the variable in multiple integrals

1. ### kidsmoker

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!

Last edited: Jan 22, 2009
2. ### Dick

25,913
The rectangle looks ok. But haven't you got the jacobian factor upside down?

3. ### kidsmoker

88
Ah yeah, should be 1/2. Other than that though does my method look correct?

Thanks.

4. ### Dick

25,913
Looks ok to me.

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