Evaluate the integral e^xy w region d xy=1, xy=4, x/y=1, x/y=2

Make a change of coordinates:




The Jacobian for this transformation is



and has a determinant of



Note that we need to use the Jacobian in the other direction; that is, we've computed



but we need the Jacobian determinant for the reverse transformation (from to . To do this, notice that



we need to take the reciprocal of the Jacobian above.

The integral then changes to

Most of the people don’t make the sense of taking it