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.

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

is a cuboid with side lengths 5, 4, and 2, so its volume is .

Meanwhile,



so the average value of over is .

The variables should add up to equal 1