Integral over Sphere Problem

Problem statement:

Find the integral of x² over (the surface of) the unit sphere in ℝ³ without doing any calculation.

Solution using calculus

Solving this with calculus isn't hard. But the problem statement says "without doing any calculation". So I haven't bothered. The symmetry solution is so much esaier anyway.

Solution using symmetry

The equation of a sphere in ℝ³ is x²+y²+z²=1. So the integral over the sphere of x²+y²+z² equals the integral over the sphere of 1, which is the area of the sphere, 4π.

So by symmetry the integral of x² is a third of that: 4π/3.

This puzzle was proposed by Nick Krempel.

This is one of several pages on using symmetry in mathematics.