## Problem Statement

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

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=4(2 \cos t-\cos 2 t) \\
y=4(2 \sin t-\sin 2 t)
\end{array}\right. \\
& 0 \leq t \leq \pi
\end{aligned}
$$
## 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:

$$
\begin{aligned}
& x=4(2 \cos t-\cos 2 t) ; x^{\prime}(t)=4(-2 \sin t+2 \sin 2 t)=8(\sin 2 t-\sin t) \\
& y=4(2 \sin t-\sin 2 t) ; y^{\prime}(t)=4(2 \cos t-2 \cos 2 t)=8(\cos t-\cos 2 t)
\end{aligned}
$$

Then, using the above formula, we get:

$$
\begin{aligned}
L & =\int_{0}^{\pi} \sqrt{(8(\sin 2 t-\sin t))^{2}+(8(\cos t-\cos 2 t))^{2}} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{\sin ^{2} 2 t-2 \sin 2 t \sin t+\sin ^{2} t+\cos ^{2} t-2 \cos 2 t \cos t+\cos ^{2} 2 t} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{1-2 \sin 2 t \sin t+1-2 \cos 2 t \cos t} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{2} \sqrt{1-\sin 2 t \sin t-\cos 2 t \cos t} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{2} \sqrt{1-\sin t \cos t \sin t-\left(1-\sin ^{2} t\right) \cos t} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{2} \sqrt{1-\sin { }^{2} t \cos t-\cos t+\sin ^{2} t \cos t} d t= \\
& =\int_{0}^{\pi} 8 \sqrt{2} \sqrt{1-\cos t} d t=\int_{0}^{\pi} 8 \sqrt{2} \sqrt{2} \sin ^{2} \frac{t}{2} d t= \\
& =\int_{0}^{\pi} 16 \sin \frac{t}{2} d t=\int_{0}^{\pi} 16 \cdot 2 \sin \frac{t}{2} d\left(\frac{t}{2}\right)=\left.32\left(-\cos \frac{t}{2}\right)\right|_{0} ^{\pi}=-32\left(\cos \frac{\pi}{2}-\cos 0\right)=-32(0-1)=32
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%BD%D0%B8%D0%BA_%D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2%D0%B0_%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_%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_18-26"

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

- Last modified: 13:58, 22 May 2010.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 18-27

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