Surface Area with Polar Coordinates — Question 9

Question 9

Problem

Find the area generated when $r=2\cos\theta$, $0\le\theta\le\pi/2$, rotates about the line $y=-1$.

Question 9 – 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 $ds=2d\theta$ and radius is $y+1=2\sin\theta\cos\theta+1$.

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