We will be looking at surface area in polar coordinates in
this section. Note however that all
we’re going to do is give the formulas for the surface area since most of these
integrals tend to be fairly difficult.

We want to find the surface area of the region found by
rotating,

about the *x* or *y*-axis.

As we did in the tangent and
arc length sections we’ll write the curve in
terms of a set of parametric equations.

If we now use the parametric formula for finding the surface
area we’ll get,

Note that because we will pick up a from the *ds*
we’ll need to substitute one of the parametric equations in for *x* or *y*
depending on the axis of rotation. This
will often mean that the integrals will be somewhat unpleasant.