## Problem Statement

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

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=3.5(2 \cos t-\cos 2 t) \\
y=3.5(2 \sin t-\sin 2 t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$
## Solution

The length of the arc of a curve defined by parametric equations is determined by the formula

$$
L=\int_{t_{1}}^{t_{2}} \sqrt{\left(x^{\prime}(t)\right)^{2}+\left(y^{\prime}(t)\right)^{2}} d t
$$

Let's find the derivatives with respect to $t$ for the given curve:

$$
\begin{aligned}
& x=3.5(2 \cos t-\cos 2 t) ; x^{\prime}(t)=3.5(-2 \sin t+2 \sin 2 t)=7(\sin 2 t-\sin t) \\
& y=3.5(2 \sin t-\sin 2 t) ; y^{\prime}(t)=3.5(2 \cos t-2 \cos 2 t)=7(\cos t-\cos 2 t)
\end{aligned}
$$

Then, using the above formula, we get:

$$
\begin{aligned}
L & =\int_{0}^{\pi / 2} \sqrt{(7(\sin 2 t-\sin t))^{2}+(7(\cos t-\cos 2 t))^{2}} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{\sin ^{2} 2 t-2 \sin 2 t \sin t+\sin ^{2} t+\cos ^{2} t-2 \cos 2 t \cos t+\cos ^{2} 2 t} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{1-2 \sin 2 t \sin t+1-2 \cos 2 t \cos t} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{2} \sqrt{1-\sin 2 t \sin t-\cos 2 t \cos t} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{2} \sqrt{1-\sin t \cos t \sin t-\left(1-\sin ^{2} t\right) \cos t} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{2} \sqrt{1-\sin { }^{2} t \cos t-\cos t+\sin ^{2} t \cos t} d t= \\
& =\int_{0}^{\pi / 2} 7 \sqrt{2} \sqrt{1-\cos t} d t=\int_{0}^{\pi / 2} 7 \sqrt{2} \sqrt{2} \sin ^{2} \frac{t}{2} d t=\int_{0}^{\pi / 2} 14 \sin \frac{t}{2} d t= \\
& =\int_{0}^{\pi / 2} 14 \cdot 2 \sin \frac{t}{2} d\left(\frac{t}{2}\right)=\left.28\left(-\cos \frac{t}{2}\right)\right|_{0} ^{\pi / 2}=-28\left(\cos \frac{\pi}{4}-\cos 0\right)=-28\left(\frac{\sqrt{2}}{2}-1\right)=14(2-\sqrt{2})
\end{aligned}
$$

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_18-14»

Categories: Kuznetsov's Problem Book Integrals Problem 18 | Integrals | Problems for Checking

- Last modified: 13:38, 22 May 2010.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 18-15

## Material from PlusPi