As before, the integral is often computed by using an iterated integral. In general it is impossible to set up an iterated integral for finding ∫ _{E}fdV for arbitrary regions, E but when the region is sufficiently simple, one can make progress. Suppose the region E over which the integral is to be taken is of the form E =
Then

It might be helpful to think of dV = dzdA. Now ∫ _{a}
Example 14.3.2 Find the volume of the region E in the first octant between z = 1 −
In this case, R is the region shown.
Thus the region E is between the plane z = 1 −

Of course iterated integrals have a life of their own although this will not be explored here. You can just write them down and go to work on them. Here are some examples.
Example 14.3.3 Find ∫ _{2}^{3} ∫ _{3}^{x} ∫ _{3y}^{x}
The inside integral yields ∫ _{3y}^{x}

Example 14.3.4 Find ∫ _{0}^{π} ∫ _{0}^{3y} ∫ _{0}^{y+z} cos
The inside integral is ∫ _{0}^{y+z} cos

Finally, this last expression must be integrated from 0 to π. Thus


Example 14.3.5 Here is an iterated integral: ∫ _{0}^{2} ∫ _{0}^{3−}
The inside integral is just a function of x and y. (In fact, only a function of x.) The order of the last two integrals must be interchanged. Thus the iterated integral which needs to be done in a different order is

As usual, it is important to draw a picture and then go from there.
Thus this double integral equals

Now substituting in for f

Example 14.3.6 Find the volume of the bounded region determined by 3y+3z = 2,x = 16−y^{2},y = 0,x = 0.
In the yz plane, the first of the following pictures corresponds to x = 0.
Therefore, the outside integrals taken with respect to z and y are of the form ∫ _{0}^{}

Example 14.3.7 Find the volume of the region determined by the intersection of the two cylinders, x^{2} + y^{2} ≤ 1 and x^{2} + z^{2} ≤ 1.
The first listed cylinder intersects the xy plane in the disk, x^{2} + y^{2} ≤ 1. What is the volume of the three dimensional region which is between this disk and the two surfaces, z =

Note that I drew no picture of the three dimensional region. If you are interested, here it is.
One of the cylinders is parallel to the z axis, x^{2} + y^{2} ≤ 1 and the other is parallel to the y axis, x^{2} + z^{2} ≤ 1. I did not need to be able to draw such a nice picture in order to work this problem. This is the key to doing these. Draw pictures in two dimensions and reason from the two dimensional pictures rather than attempt to wax artistic and consider all three dimensions at once. These problems are hard enough without making them even harder by attempting to be an artist.