Surface Area with Polar Coordinates — Question 4

Question 4

Problem

Why must $|r\sin\theta|$ replace $r\sin\theta$ when a polar curve crosses the $x$-axis during rotation about that axis?

Question 4 – 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. Radius to the axis is the nonnegative distance $|y|=|r\sin\theta|$.

4. Without the absolute value, areas on opposite sides would cancel.