Arc Length and Surface Area Revisited — Question 7

Question 7 – 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. For radius $R>0$ and height $h>0$, rotate $(x,y)=(Rt/h,h-t)$, $0\le t\le h$, about the $y$-axis.

4. Then $ds=(\sqrt{R^2+h^2}/h)dt$ and $S=\pi R\sqrt{R^2+h^2}=\pi R\ell$, matching the mnemonic.