Surface Area with Polar Coordinates — Question 5

Question 5 – 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. The interval $0\le\theta\le\pi/2$ traces only the right semicircle; rotation preserves the $x$-coordinate, so the left half is not duplicated.

4. Use $\boxed{0\le\theta\le\pi}$ to trace the full generating circle once.