Surface Area with Polar Coordinates — Question 8

Question 8

Problem

A curve is entirely in the first quadrant. Compare its areas of revolution about the $x$- and $y$-axes using only weighted average coordinates.

Question 8 – 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. $S_x=2\pi L\bar y_s$ and $S_y=2\pi L\bar x_s$, where the bars are arc-length-weighted means.

4. Therefore $S_x/S_y=\bar y_s/\bar x_s$.