Surface Area with Polar Coordinates — Question 6

Question 6

Problem

For $r=R$, $0\le\theta\le\pi$, rotate the semicircle about the $x$-axis and recover the sphere area.

Question 6 – 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. $y=R\sin\theta$ and $ds=R\,d\theta$.

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