INSTRUCTOR: Find the average value of the function f of x comma y comma z, equal to x times y times z over the cube, with sides of length 4 units in the first octet, with one vertex at the origin and edges parallel to the coordinate axes. In order to find the average value, we're going to find the triple integral of this function over the given region and then divide that by the area of the region itself. I say area, when in reality, it's average value. So it's the volume because we're dealing with, our region is a cube.

So the area of, I'm going to call it v, is going to be equal to 4 cubed, which is 64. So we'll hold on to that value. Now, for the triple integral, we'll set this up as the integral from 0 to 4 with respect to x of the integral from 0 to 4 with respect to y of the integral from 0 to 4 with respect to z of the function x times y times z. Because these are three independent functions in x, y, and z respectively, we can rewrite this as the product of 3 integrals.

So the integral from 0 to 4 with respect to x of the function x, I'm going to put that in parentheses, multiplied by parentheses the integral from 0 to 4 with respect to y of the function y close parentheses, multiplied by the integral from 0 to 4 with respect to z of the function z. That's also in parentheses. Now multiply all three of those. The value of each of those is 8.

So this is equal to 8 times 8 times 8, so that our triple integral is equal to 512. Now, for the average value, again, we will take this outcome, 512, divided by the-- I mislabeled that earlier. I said a of v. But really, that's also saying area of the region.

So we just know that region that's not actually labeled here. We know that it has a volume of 63. So we're going to take our 512, or 64, rather. We are going to take our 512 for f average, 512 divided by 64. So that the average value over that region is 8.