## Problem Statement

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

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

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

We get:

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

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%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_18-3»

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

Kuznetsov's Problem Book Integrals 18-3 — PlusPi

Kuznetsov's Problem Book Integrals 18-3 — PlusPi
Created by GeeTeatoo

(

![](https://cdn.mathpix.com/cropped/2024_05_22_b045d7f8347528b5fcdfg-05.jpg?height=52&width=45&top_left_y=-3&top_left_x=2042)$\qquad$

Correct

![](https://cdn.mathpix.com/cropped/2024_05_22_b045d7f8347528b5fcdfg-05.jpg?height=91&width=46&top_left_y=263&top_left_x=999)

![](https://cdn.mathpix.com/cropped/2024_05_22_b045d7f8347528b5fcdfg-05.jpg?height=71&width=60&top_left_y=266&top_left_x=1105)

Is

ser

Year

.
-

$\square$

![](https://cdn.mathpix.com/cropped/2024_05_22_b045d7f8347528b5fcdfg-05.jpg?height=63&width=60&top_left_y=1328&top_left_x=1191)
-

Is

(

## Kuznetsov's Problem Book Integrals 18-4

## Material from PlusPi