How do you use an arc length integral to show the length of the circle of radius r?

The integral from 0 to 2 pi of your constant value r will equal the circumference. This will be equal to 2*pi*r.

This can be derived because of the following:

Arc length = integral from a to b of sqrt(r^2-(dr/dtheta)^2) dtheta. By substituting the equation r = a constant c, dr/dtheta will equal 0, a will equal 0, and b will equal 2pi (the radians in a circle).

By substitution, this becomes the integral from 0 to 2 pi of sqrt(c^2 + 0)dtheta, which leads us back to the original formula.