INSTRUCTOR: Consider the solid sphere E equal to the set of all points x comma y comma z, such that x squared plus y squared plus z squared equals 9. Write the triple integral over the region E of the function f of x comma y comma z for an arbitrary function f as an iterated integral. Then evaluate this triple integral with f of x comma y comma z equal to 1.

Notice that this gives the volume of a sphere using a triple integral. With an arbitrary function f, our boundaries are as follows. This interval will be integral from negative 3 to 3 with respect to x of the integral from negative square root of 9 minus x squared, end square root, to square root of 9 minus x squared, end square root, with respect to y of the integral from negative square root of 9 minus x squared minus y squared end square root, to square root of 9 minus x squared minus y squared, end square root, with respect to z. And of course, the function here is f of x comma y comma z.

Now, we will evaluate this with our function equal to 1. So the first layer of that, integrating the first layer of the innermost function there, this becomes the integral from negative 3 to 3 with respect to x of the integral from negative 9 minus x squared, end square root, to square root of 9 minus x squared, end square root, of 2, that's with respect to y, of the function 2 square root of 9 minus x squared minus y squared, end square root. Now, applying trig substitution methods to this next layer because it's a square root function, this becomes the integral from negative 3 to 3 of negative pi open parenthesis x squared minus 9 close parentheses. That's with respect to x. And this, when we find the antiderivative and evaluate at both endpoints, we come up with a value of 36 pi.

Now, that value is interesting because as the question stated, notice this gives the volume of a sphere using a triple integral. The function of the boundary we began with, x squared plus y squared plus z squared equals 9, is a sphere with a radius of 3. So applying some geometry, that tells us the volume would be 4/3 pi multiplied by 3 cubed, which is, in fact, 36 pi.