Change of variables in triple integration using polar coordinates The World Geodetic System (the international standard used by GPS devices) models the Earth as an oblate spheroid described by the equation $$x^2/a^2+y^2/a^2+z^2/b^2=1$$ where a ≈ 6378.1370 km is the equatorial radius, and b ≈ 6356.7523 km is the polar radius. Use this information and a triple integral to calculate an estimate of the volume of the Earth.

The triple integral for the volume is

$$V=\int_{A(x,y)} \int_{-b\sqrt{1-x^2/a^2-y^2/a^2}}^{b\sqrt{1-x^2/a^2-y^2/a^2}}dzdydx$$ $$=2b\int_{A(x,y)} \sqrt{1-x^2/a^2-y^2/a^2}dydx$$

In polar coordinates,

$$V= 2b\int_0^{2\pi}\int_0^a \sqrt{1-r^2/a^2}rdrd\theta=\frac43\pi ba^2$$

which is about 1.083$\times 10^{12}$km$^3$.