Question

The arc length of polar functions: Find the arc lengths of the following polar functions.$r=a\cos \theta$

, where is a positive constant, for

Short answer

The arc length is$2a\mathrm{\pi }\mathrm{units}$

Step-by-step solution

Given information.

$r=a\cos \theta$and$\theta \in \left[0,2\mathrm{\pi }\right]$

Arc length.

The length of an arc is given by,

$L={\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta$

So,

role="math" localid="1662989532460" $$\begin{gathered} L={\int }_0^{2\mathrm{\pi }}\sqrt{a^2{\cos }^2\theta +{\left(-a\sin \theta \right)}^2}d\theta \left[\because \frac{dr}{d\theta }=-a\sin \theta \right] \\ ={\int }_0^{2\mathrm{\pi }}\sqrt{a^2{\cos }^2\theta +a^2{\sin }^2}\theta d\theta \\ ={\int }_0^{2\mathrm{\pi }}ad\theta \left[\because {\cos }^2\theta +{\sin }^2\theta =1\right] \\ =2a\mathrm{\pi }-0 \\ =2a\mathrm{\pi }\mathrm{units} \end{gathered}$$

Thus, the arc length is$2a\mathrm{\pi }\mathrm{units}$