Triple Integrals — Question 5

Question 5

Find the average of $f=x^2+y^2+z^2$ on $[-1,1]^3$.

Tasks

1. Use symmetry and separability.

2. Compute the average.

3. Compare with the range.

Question 5 – Solution

Strategy. The three squared-coordinate contributions are identical.

Step 1: Integral$\iiint_Bx^2dV=(\int_{-1}^1x^2dx)(2)(2)=8/3$. Thus $\iiint_Bf\,dV=3(8/3)=8$.

Step 2: AverageSince $\operatorname{vol}(B)=8$, $\boxed{f_{\mathrm{avg}}=1}$.

Verification$0\le f\le 3$, and each coordinate square has average $1/3$, so their sum has average $1$.