## Problem Statement

Calculate the lengths of the arcs of the curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=3(\cos t+t \sin t) \\
y=3(\sin t-t \cos t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{3}
\end{aligned}
$$
## Solution

$\left\{\begin{array}{l}x=3(\cos t+t \sin t) \\ y=3(\sin t-t \cos t)\end{array}\right.$

$x^{\prime}(t)=3(-\sin t+\sin t+t \cos t)=3 t \cos t$

$y^{\prime}(t)=3(\cos t-\cos t+t \sin t)=3 t \sin t$

$$
L=\int_{0}^{\frac{\pi}{3}} \sqrt{9 t^{2} \cos ^{2} t+9 t^{2} \sin ^{2} t} d t=\quad 3 \int_{0}^{\frac{\pi}{3}} t d t=\left.\frac{3}{2} t^{2}\right|_{0} ^{\frac{\pi}{3}}=\frac{\pi^{2}}{6}
$$

Source — "http://pluspi.org/wiki/index.php/?\?\?\?\?\?\?\?\?\?\?\?_ \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_18-9$ "

Categories: Kuznetsov's Problem Book Integrals Problem 18 | Integrals | Popular Problems

- Last edited on this page: 13:07, 24 April 2010.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 18-10

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