Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 3 / 9C / Solution

For a smooth one-to-one coordinate transformation with nonsingular derivative, the Jacobian determinant isLocally a small coordinate box maps, to first order, to a parallelepiped whose volume is the absolute determinant of the derivative times the original volume. Summing these local volume approximations gives the change of variables formula, with $|J|$ accounting for either orientation. The usual regularity and nonsingularity hypotheses are part of this substitution theorem.

The region is the upper half of the spherical shell between radii two and three, including its annular flat boundary in the equatorial plane. Use spherical coordinates $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$ with $2\le r\le 3$, $0\le \theta \le \pi /2$, $0\le \varphi <2\pi$. Their Jacobian determinant is $r^2\sin \theta$; the polar axis and azimuth seam have measure zero, so do not obstruct this integration.

The integrand is $x^2+y^2=r^2{\sin }^2\theta$. Consequently