Arc Length and Surface Area Revisited — Question 8

Question 8

Problem

A curve has polar form $r=f(\theta)$ but is rotated about the $y$-axis. State when converting to Cartesian form might be safer.

Question 8 – Solution

Solution

1. Choose a representation and interval that trace the desired curve exactly once. The equivalent arc-length formulas are $$\begin{aligned} L&=\int\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx,\\ L&=\int\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2}\,dt,\\ L&=\int\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta. \end{aligned}$$

2. For a surface of revolution, multiply the appropriate arc-length element by $2\pi$ times the nonnegative distance to the axis. Check the tracing interval to prevent geometric double-counting.

3. Conversion is safer when $x=r\cos\theta$ changes sign or the polar interval retraces points, because the radius to the $y$-axis is $|x|$ and double-covering is easier to see geometrically.