## Problem Statement

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

$$
\rho=5 \varphi, 0 \leq \varphi \leq \frac{12}{5}
$$
## Solution

The length of the arc of a curve given by an equation in polar coordinates is determined by the formula

$$
L=\int_{\varphi_{0}}^{\varphi_{1}} \sqrt{(\rho(\varphi))^{2}+\left(\rho^{\prime}(\varphi)\right)^{2}} d \varphi
$$

For the curve given by the equation $\rho=5 \varphi$, we find: $\rho^{\prime}=5$

We get:

$$
\begin{aligned}
L & =\int_{0}^{12 / 5} \sqrt{(5 \varphi)^{2}+5^{2}} d \varphi= \\
& =5 \int_{0}^{12 / 5} \sqrt{\varphi^{2}+1} d \varphi={ }^{(1)} \\
& =5\left(\left.\frac{\varphi}{2} \sqrt{\varphi^{2}+1}\right|_{0} ^{12 / 5}+\left.\frac{1}{2} \ln \left|\varphi+\sqrt{\varphi^{2}+1}\right|\right|_{0} ^{12 / 5}\right)= \\
& =5\left(\frac{1}{2} \cdot \frac{12}{5} \sqrt{\frac{144}{25}+1}-0\right)+5\left(\frac{1}{2} \ln \left|\frac{12}{5}+\sqrt{\frac{144}{25}+1}\right|-\ln |0+\sqrt{0+1}|\right)= \\
& =6 \cdot \sqrt{\frac{169}{25}}+5\left(\frac{1}{2} \ln \left|\frac{12}{5}+\sqrt{\frac{169}{25}}\right|-0\right)= \\
& =6 \cdot \frac{13}{5}+\frac{5}{2} \ln \left|\frac{12}{5}+\frac{13}{5}\right|=\frac{78}{5}+\frac{5}{2} \ln 5
\end{aligned}
$$

In (1) we used the formula: $\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}+a^{2}}\right|$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� $\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{\%} \% \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 1 \% 82$ $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+19-25$ "

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

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