On the change of variables in the multiple integrals.

In Calculus I, a useful technique to evaluate many difficult integrals is by using a u-substitution, which is essentially a change of variable to simplify the integral. Sometimes changing variables can make a huge difference in evaluating a double integral as well, as we have seen already with polar coordinates. This is often a helpful technique for triple integrals as well. In general, say that we have a transformation T pu, vq “ px, yq that maps a region S to a region R (see picture below). All images are taken from Stewart, 8th Edition.