Limits in Double Integration

can someone please help me understand how to find the 'new' limits for double integration. I know that you have to split up the area and fix x or y. If you can go through an example with me then I would be grateful as I keep getting this wrong. We have only been taught how to work out the area of a triangle and so a quadrilateral makes no sense to me.... Btw please don’t give 'genera;' advice because I seriously won’t get it. I have looked in my books/lecture notes but it’s all just general theory which isn’t helpful.

Example question: By making an appropriate substitution, find the area of D and check the substitution is a 1-1 transformation by finding the inverse explicitly.

$$ \iint_D \exp{[xy(x-y)]} (x^2-y^2) {\rm d}x\, {\rm d}y$$

Here is the diagram: enter image description here Thanks for the help!

Solution 1:

You don't have to split up the area. The boundary curves of $D$ are given by constant values of $x-y$ and $xy$. Thus, it makes sense to transform to new variables $u=x-y$ and $v=xy$; then the integration region will turn into an axis-parallel rectangle, and you can just perform the integration without further worrying about the integration limits.

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