Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 3 / 10C (Vector Calculus)

For a one-to-one continuously differentiable change of variables with nonzero Jacobian determinant in the interior, the change of variables formula isThe absolute value is necessary even if the transformation reverses orientation.

The original region is the first-quadrant area between the hyperbola and circle, above the horizontal-axis segment. More explicitly,Its vertices are $(1,0)$, $(\sqrt{2},0)$ and $(\sqrt{3/2},1/\sqrt{2})$. Under the squared-radius difference substitution, the inverse isand the image is the triangle $1\le v\le u\le 2$, with vertices $(1,1),(2,1),(2,2)$.

The forward Jacobian determinant isIt is nonzero in the interior; the zero on $y=0$ lies on a measure-zero boundary, so the integral formula applies by taking interior limits. Since $x^{5}y-x{y}^5=xyuv$ and $4x^2{y}^2=u^2-v^2$, the inverse-Jacobian factor cancels $xy$, givingIntegrating in $v$ first and then $u$,Therefore