Solving Double Integrals with Change of Variables

[Zaphodx57x]

Does anyone know of any sources that explain change of variables for double integrals. Actually, I get the change of variables thing, but a few of our problems don't give us the transforms. I don't understand how to create these myself.

Here is an example:
Math Problem

So far, I found all the x,y coordinates of the joints because I know these joints or cross sections will exist after we change variables. However, I don't know where to go from there. Can I essentially make any transform I like?

 

[arildno]

1) Sources:
In my opinion, an excellent intuitive understanding of the change of variables stuff can be found in Marsden&Tromba "Vector Calculus" (Lots of editions..).
Instead of "burdening" the reader with rigorous proofs of the change-of-variables theorem, it has a clear focus on how to generate the "proper" area elements dA. (Lots of worked examples as well!)

If you want a more rigorous treatment, one book is Marsden "Introduction to Real Analysis"

2) Problem:
Note that your region is bounded by two pairs of parallell lines.
Try setting u=2x-y, v=3x+y, and see what you get.

 

[Zaphodx57x]

2) Problem:
Note that your region is bounded by two pairs of parallell lines.
Try setting u=2x-y, v=3x+y, and see what you get.

Wow, thank you. You just pointed out a valuable thing to me. I think I "get" what I'm supposed to do for these problems now. Thank you.

 

[Divergent13]

Yes that is the way to go--- I just consider C o V as the "U-substitution" chapter but for Double Integrals. (Or even triple--- but i hate calculating 3x3 determinants!)