Surface Area with Polar Coordinates — Question 1

Question 1

Problem

The upper semicircle $r=2R\cos\theta$, $0\le\theta\le\pi/2$, rotates about the polar axis. Find the generated surface area.

Question 1 – 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. Here $y=2R\sin\theta\cos\theta$ and $ds=2R\,d\theta$.

4. Thus $S=8\pi R^2\int_0^{\pi/2}\sin\theta\cos\theta d\theta=\boxed{4\pi R^2}.$