Question

Use a triple integral in spherical coordinates to derive the volume of a sphere with radius a.

Answer 1

Solution

In cartessian coordinates dV = dxdydz. In spherical coordinates dV = r²sinϕdrdϕdϴ (r≥0, 0≤ϕ≤π).

$V = \iiint_VdV = \int_0^a\int_0^\pi\int_0^{2\pi}r^2sin\phi d\theta d\phi dr = \int_0^ar^2dr \int_0^\pi sin\phi d\phi \int_0^{2\pi} d\theta =$

$=\frac{1}{3}a^3 (-cos\pi+cos0)2\pi = \frac{4\pi}{3}a^3$