Arc Length and Surface Area Revisited — Question 1

Question 1 – 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. Cartesian: $\int_0^4\sqrt{1+9x/4}dx$.

4. Parametric with $0\le t\le2$: $\int_0^2t\sqrt{4+9t^2}dt$.

5. Substitution $x=t^2$ makes them identical.