This is a multiple-choice question from an old exam, meaning it should be answerable in minutes without writing out anything. I'm afraid therefore that I'm missing some crucial insight/intuition about integrals.
Let $M$ be the surface of the unit sphere in $\mathbb R^n$ and $\omega_M=\int_M1d\sigma$. Which is true?
- $\int_Mx_ix_jd\sigma = 0, 1\le i, j\le n$
- $\int_Mx_ix_jd\sigma = \frac 1 {n^2}\omega_M, 1\le i, j\le n$
- $\int_Mx_ix_jd\sigma = 0, 1\le i\ne j\le n$
- $\int_Mx_i^2d\sigma=\frac 1 {n^2}\omega_M, 1\le i\le n$
I could disprove options 1. and 2. in my head and in fact 3. and 4. hold in $\mathbb R^2$ for the Lebesgue measure, i.e., $\sigma=\lambda$.
Is there some semi-obvious reason why 3. or 4. is correct?