We use integrals to model mass.

### Mass

We learned some time ago that if the density of an object is uniform, When the density of an object is not uniform, we define a density function and mass , to write: Summing these together with an integral, we find the mass is equal to

### Moments and center of mass

A *moment* is a scalar quantity describing how mass is distributed in relation to a
point, line, or plane.

- The
**moment**about the -plane is given by - The
**moment**about the -plane is given by - The
**moment**about the -plane is given by

The moments are directly related to the center of mass of an object.

**center of mass**is defined to be the point

*four*triple integrals. Each computation will require a number of careful steps. Get out several sheets of paper and take a deep breath. First we’ll compute the mass. Write with me:

Now we’ll compute . Write with me:

Now we’ll compute . Write with me:

Now we’ll compute . Write with me:

The center of mass is

As stated before, there are many uses for triple integration beyond finding volume. When describes a rate of change function over some space region , then gives the total change over . Our example of this was computing mass via a density function. Here a density function is simply a “rate of mass change per volume” function. Thus, integrating density gives total mass.

While knowing *how to integrate* is important, it is arguably much more important to
know *how to set up* integrals. It takes skill to create a formula that describes a
desired quantity; modern technology is very useful in evaluating these formulas
quickly and accurately.