Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 3 / 10A (Vector Calculus)

Choose coordinates that measure the curved boundary and the slope of a ray:On the positive interior, the inverse change of variables isThe two rays become $v=a$ and $v=1$, while the curve becomes $u=1$. The interior of $S$ therefore maps to the rectangle $0<u<1$, $a<v<1$. For $a=0$, the lower edge $v=0$ describes a limiting boundary, since the original region is unbounded; the origin also has no uniquely defined ratio $y/x$. For $a=1$ the region has zero area. These endpoint cases do not invalidate the change of variables on the positive interior.

For the three-dimensional integral, retain $z$ as the third coordinate. The transformed region isThe planar Jacobian determinant isThe integrand cancels this Jacobian determinant especially simply:HenceOne can first use the change of variables formula with $z\ge \epsilon >0$ and then pass to the limit. The nonnegative integrand allows this by the monotone convergence theorem, so the unbounded limiting slice at $z=0$ causes no omitted contribution.