Surface Area with Polar Coordinates — Question 7

Question 7 – 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. Use radius $|x|=|r\cos\theta|$ and $ds=\sqrt{\cos^22\theta+4\sin^22\theta}d\theta$.

4. On this interval $r\cos\theta\ge0$, so the requested setup is $$\boxed{S=2\pi\int_{-\pi/4}^{\pi/4}\cos2\theta\cos\theta \sqrt{\cos^22\theta+4\sin^22\theta}\,d\theta}.$$