How do you compute an average location?

Well this starts off simple enough.

If I say, what's the average of two points?

Well, one draws the line segment and picks the point in the middle.

You could probably use some basic geometry to determine the average

of three locations, or the average of four locations.

But what would you do if asked to compute the average of an entire region?

This requires thinking in terms of calculus.

Breaking that region up into infinitesimal elements and

then computing an average, and that's exactly what we'll do in defining

the centroid is the average location in a domain, let's call that D.

In this case, if we set up x and y coordinates,

then we would characterize the centroid in terms of it's coordinates,

x r and y bar.

That notation is chosen to help you remember the definition.

X bar is the average x coordinates over the domain.

That is the integral of x over d divided by the integral of one over d.

Y bar correspondingly is the integral of y over D

divided by the integral of 1 over D.

In higher dimensional settings, I bet you can guess what the formula is.

We're going to use the perspectives from the bonus materials

in lecture 31 to compute these integrals.

Dividing the region not into strips, but

rather into infinitesimal rectangles of dimensions dx and

dy and then averaging over those.

The notation that is most useful to us is that of

a double integral, or an iterated integral.

The denominator in both of these cases, being the integral of one over the domain,

is really the integral with respect to the area element.

If you integrate dA, what do you get?

Well you get of course A, so one way to write these formula for

the centroid is as 1 over A times the integral over d

of x dA, or of y dA respectively.