Finally we must write in terms of . Unfortunately, from our inequalities above, there is no direct way to get this. We must think about what our solid looks like. Recall that the plane bounding the solid is . If , then our plane is the line . Hence We may now write our desired integral:

We study integrals over general regions by integrating .

**integrand**is the “thing” you are integrating: Think about this: The integral computes the length between and . This is because the area between the constant curve of height one and the -axis is just the length of the curve! In a similar way: Here you imagine a surface of constant height above a region :

### Double integrals and area

We start with a third version of Fubini’s Theorem.

- If
- If

It is important to note that when using Fubini’s Theorem, we must always have numbers as the limits of the outer-most integral and curves (note constant curves are numbers) as the limits of the inner-most integral: Note, if we set and (or and ) to be constant functions, we recover the version of Fubini’s Theorem from the previous section.

Whenever you learn a new technique, you should always “try it out” on a computation where you know the answer through a different method. So let’s used an integral to find the area of a triangle.

#### Changing the order of integration

We can change the order of integration. This can make a big difference in the difficulty of the integral in question. This can be somewhat challenging. If you have a graph of the region in question, this can help quite a bit. If not, then a strategy is to

- Find absolute bounds for each variable.
- Solve inequalities that will allow you to integrate:

Let’s see an example.

Let’s see another example.

Sometimes when you switch the order of integration, you will need to write the sum of iterated integrals. You can recognize this by the fact that you won’t be able to find a single curve to bound your inner integral. This will be more clear with an example.

Now let’s integrate with respect to and then . If we give absolute bounds to our inequalites
above, we see something strange: This is actually **two** inequalities (note we can see this
from the graph):

**different**curves! When this happens, it is probably best to look at the graph of the situation:

#### Why change the order of integration?

You may ask yourself, “Why change the order of integration?” Sometimes changing the region can make a difficult (impossible) antiderivative easier (not impossible). Let’s see an example.

This author does not know how to solve the problem above without changing the order of integration.

### Triple integrals and volume

We start by introducing a fourth version of Fubini’s Theorem.

*six*reorderings total with three variables. We will spare the young mathematician the details, and trust that you will sort it out.

Again note that when using Fubini’s Theorem, we must always have numbers as the limits of the outer-most integral, curves as the limits of the middle integral, and surfaces as the limits of the inner-most integral:

Now with Fubini’s help, we will use triple integrals to compute volumes.

- The plane .
- The plane .
- The plane .
- The plane .