Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 12C / i / Solution

The inequalities $y\le x\le 2y$ force $y\ge 0$, and the reciprocal bounds exclude zero. Thus the whole region lies in the positive quadrant. Introduce ratio-product coordinatesThe region becomes the rectangle $1\le u\le 2$, $1\le v\le 2$. The inverse map and its Jacobian determinant areThe change of variables formula then gives