## Problem Statement

Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.

$$
\rho=6 \sin \varphi, 0 \leq \varphi \leq \frac{\pi}{3}
$$
## Solution

$$
\begin{aligned}
L & =\int_{0}^{\pi / 3} \sqrt{\rho^{2}+\left(\rho^{\prime}\right)^{2}} d x \\
\rho^{\prime} & =6(\sin \varphi)^{\prime}=6 \cos \varphi \\
L & =\int_{0}^{\pi / 3} \sqrt{(6 \sin \varphi)^{2}+(6 \cos \varphi)^{2}} d \varphi=6 \int_{0}^{\pi / 3} \sqrt{(\sin \varphi)^{2}+(\cos \varphi)^{2}} d \varphi=6 \int_{0}^{\pi / 3} d \varphi=\left.6 \varphi\right|_{0} ^{\pi / 3}=6(\pi / 3)-6 \cdot 0=2 \pi
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\�\�\�\�\� $\% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB}$ $\% \mathrm{D} 1 \% 8 \mathrm{~B} \_19-31 »"

Categories: Kuznetsov's Problem Book Integrals Problem 19 | Integrals

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