The average value of a function [tex]f[/tex] over some domain [tex]\mathcal D[/tex] is the ratio of the integral of [tex]f[/tex] over [tex]\mathcal D[/tex] to the measure of [tex]\mathcal D[/tex]. In 3 dimensions, you're dividing the integral of [tex]f[/tex] by the volume of [tex]\mathcal D[/tex].
[tex]\mathcal D[/tex] is a cuboid with side lengths 5, 4, and 2, so its volume is [tex]5\cdot4\cdot2=40[/tex].
Meanwhile,
[tex]\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[/tex]
so the average value of [tex]f(x,y,z)[/tex] over [tex]\mathcal D[/tex] is [tex]\dfrac{600}{40}=15[/tex].
