Arc Length and Surface Area Revisited — Question 5

Question 5

Problem

A noncircular ellipse lacks an elementary perimeter formula. Explain why switching among Cartesian, parametric, and polar forms does not remove the essential difficulty.

Question 5 – 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. Each representation reduces to an elliptic integral.

4. Reparametrization changes appearance, not the intrinsic arc-length integral or its non-elementary nature.