INSTRUCTOR: Evaluate the triple integral over the region B of the function z multiply by sine of x, multiplied by cosine of y, where B is equal to the set of all points x,y,z, such that 0 is less than or equal to x, is less than or equal to pi. 3 pi over 2 is less than or equal to y, is less than or equal to 2 pi. And 1 is less than or equal to z, is less than or equal to 3. First let's set up this triple integral as the integral from 0 to pi, with respect to x of the integral from 3 pi over 2 to 2 pi, with respect to y of the integral from 1 to 3, with respect to z of the function z sine x cosine y.

Now, it's in our favor here that these are three independent functions of x, y, and z because, in that case, we can rewrite this as a product of 3 separate intervals. So we can write this as the integral from 0 to pi with respect to x of the function that's in terms of x, that is sine x, putting that integral in parentheses, multiplied by big parentheses, the integral from 3 pi over 2, to 2 pi with respect to y of cosine y, close parentheses, multiplied by the integral from 1 to 3, with respect to z of the function z. Finding these each separately is not too difficult.

So the first integral is a value equal to 2. The second has a value of 1. And the third has a value of 4. So multiplying those three together, that tells us that this triple integral is equal to a value of 8.