Emmy Dillon

2022-06-19

What is the arc length of $f\left(x\right)=\sqrt{4-{x}^{2}}$ on $x\in \left[-2,2\right]$?

Nia Molina

$\frac{dy}{dx}=\frac{-x}{\sqrt{4-{x}^{2}}}$
$\left(\frac{dy}{dx}{\right)}^{2}=\frac{{x}^{2}}{4-{x}^{2}}$
arc length $=\int \sqrt{1+\frac{{x}^{2}}{4-{x}^{2}}}dx=2\int \frac{1}{\sqrt{4-{x}^{2}}}dx$
$=\mathrm{arcsin}\left(\frac{x}{2}\right)+C$
Finally, evaluate over [-2,2]
arc length $=2\pi$

Do you have a similar question?