Notions of Triple Integrals

 Objectives

1) Recognize when a triple integral is integrable over a rectangular box

2) Evaluate a triple integral using an iterated integral

Integrable  functions of three variables

We are going to follow the same procedure as we did in double integrals.  We divide the interval [a,b} into l

the following figure:

As we partition the box into into smaller and smaller boxes, the number of small boxes becomes infinitely large. In this case the triple sum tends to a number called the triple integral of the function f(x,y,z).

Definition

Theorem

Now that we define the triple integral, the Fubini's theorem allows to evaluate it the same way it was used in the case of double integral.

Example I

Solution

The order of integration is known. Let's integrate with respect to x first then y and z.

Example II

Solution

Now let's integrate in a different order to see if we get the same answer. Let's integrate with respect to x, then z and y.

162

Practice