## Problem Statement

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

$$
\rho=2 \sin \varphi, 0 \leq \varphi \leq \frac{\pi}{6}
$$
## Solution

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

Source — "http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\�\�\�\�\� \%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2 \%D0\%98\%D0\%BD\%D1\%82\%D0\%B5\%D0\%B3\%D1\%80\%D0\%B0\%D0\%BB $\%$ D1\%8B_19-29"

Categories: Kuznetsov Problem Book Integrals Problem $19 \mid$ Integrals

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## Problem Kuznetsov Integrals 19-30

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