Q. 24 - Calculus

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 

Step-by-Step Solution

Answer

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

Step 1: Given information.

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

Step 2: 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,

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

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