Surface Area with Polar Coordinates — Question 2

Question 2

Problem

Set up the surface area formed by rotating the cardioid $r=1+\cos\theta$, $0\le\theta\le\pi$, about the polar axis.

Question 2 – Solution

Solution

1. Compute the polar arc-length element $$ds=\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.$$

2. Express the radius of rotation as a nonnegative distance: use $|r\sin\theta|$ for the $x$-axis and $|r\cos\theta|$ for the $y$-axis. Then apply $$S=2\pi\int_a^b(\text{radius to the axis})\,ds,$$ over an interval that generates the surface exactly once.

3. Use $y=r\sin\theta$ and $ds=2\cos(\theta/2)\,d\theta$ on this interval.

4. Therefore $$\boxed{S=4\pi\int_0^\pi(1+\cos\theta)\sin\theta\cos(\theta/2)\,d\theta}.$$