Surface Area with Polar Coordinates — Question 3

Question 3

Problem

Rotate the spiral arc $r=\theta$, $0\le\theta\le\pi/2$, about the $y$-axis. Write the exact surface-area integral.

Question 3 – 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. Distance to the $y$-axis is $x=\theta\cos\theta$, and $ds=\sqrt{1+\theta^2}d\theta$.

4. Thus $\boxed{S=2\pi\int_0^{\pi/2}\theta\cos\theta\sqrt{1+\theta^2}d\theta}$.