## Problem Statement

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

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=e^{t}(\cos t+\sin t) \\
y=e^{t}(\cos t-\sin t)
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \pi
\end{aligned}
$$
## Solution

$$
L=\int_{\alpha}^{\beta} \sqrt{\left(x_{t}^{\prime}\right)^{2}+\left(y_{t}^{\prime}\right)^{2}} d t
$$

Find the derivatives with respect to $t$:

$$
\begin{aligned}
& x_{t}^{\prime}=e^{t}(\cos t+\sin t)+(\cos t-\sin t) e^{t}=2 e^{t} \cos t \\
& y_{t}^{\prime}=e^{t}(\cos t-\sin t)+(-\cos t-\sin t) e^{t}=-2 e^{t} \sin t \\
& L=\int_{\frac{\pi}{2}}^{\pi} \sqrt{4 e^{2 t} \cos ^{2} t+4 e^{2 t} \sin ^{2} t} d t=\int_{\frac{\pi}{2}}^{\pi} 2 e^{t} d t=2 \int_{\frac{\pi}{2}}^{\pi} e^{t} d t=\left.2 e^{t}\right|_{\frac{\pi}{2}} ^{\pi}=2\left(e^{\pi}-e^{\frac{\pi}{2}}\right)
\end{aligned}
$$

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}+18-12 »

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

- Last modified: 13:05, 9 May 2009.
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## Problem Kuznetsov Integrals 18-13

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