I'm not quite sure how to go about finding the new region of integration.
The evaluation of an integral over a region D can be done by changing variables and integrating over a new region and including the jacobian
$$\iint_Df(x,y)dxdy =\iint_{D^*}\|\frac{\delta (x,y)}{\delta (u,v)}\|f(x(u,v),y(u,v)dudv$$ But how exactly do we find $D^*$? For simpler regions like parallelograms I am able to do this geometrically usually by transforming the corner points and connecting them, but this doesnt work for non linear transformations?
Also is it always true that boundary points in $D$ map to boundary points in $D^*$?
Thanks for any help!
