Question

Find the average value of the function f \left( x, y, z \right) = x^{2} + y^{2} + z^{2} over the rectangular prism 0 \leq x \leq 5, \ \ 0 \leq y \leq 4, \ \ 0 \leq z \leq 2.

Answer 1

The average value of a function $f$ over some domain $\mathcal D$ is the ratio of the integral of $f$ over $\mathcal D$ to the measure of $\mathcal D$. In 3 dimensions, you're dividing the integral of $f$ by the volume of $\mathcal D$.

$\mathcal D$ is a cuboid with side lengths 5, 4, and 2, so its volume is $5\cdot4\cdot2=40$.

Meanwhile,

$\displaystyle\iiint_{\mathcal D}f(x,y,z)\,\mathrm dV=\int_{z=0}^{z=2}\int_{y=0}^{y=4}\int_{x=0}^{x=5}(x^2+y^2+z^2)\,\mathrm dx\,\mathrm dy\,\mathrm dz=600$

so the average value of $f(x,y,z)$ over $\mathcal D$ is $\dfrac{600}{40}=15$.