Examples of Double Integrals using Transformations
The idea behind solving double integrals using transformations is that you're usually given the transformation to apply in the question. You are usually given a shape for a region of integration, generally triangles, parallelograms, and ellipses, which we have examples of.
In general, you need to find equations that make up the shape, convert them to the transformation coordinates, set up new integral boundaries and find the Jacobian to multiply by.
This sounds a bit wordy, but it's much easier to view examples of solved problems. We have many worked-out examples where double integrals are solved using transformations below: