Changing the variable in multiple integrals

[kidsmoker]

Homework Statement

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).

Homework Equations

I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv

where J is the Jacobian.

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!

 

[Dick]

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

 

[kidsmoker]

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

Thanks.

 

[Dick]

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

Thanks.

Looks ok to me.