## Problem Statement

Calculate the volumes of solids formed by rotating figures bounded by the graphs of functions. The axis of rotation is $O x$.

$$
y=5 \cos x, y=\cos x, x=0, x \geq 0
$$
## Solution

## Since

the $O x$ 

axis 

is 

the 

axis 

of 

rotation, the volume 

is 

given 

by 

the 

formula:

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-07.jpg?height=1507&width=1177&top_left_y=994&top_left_x=568)

$$
V=\pi \cdot \int_{a}^{b} y^{2} d x
$$

Let's find the limits of integration: From the problem statement, we already have: $x_{1}=0$ Now let's find the upper limit:

$$
\cos x=5 \cos x \Rightarrow x_{2}=\frac{\pi}{2}
$$

We will find the volume of the solid as the difference in volumes of two solids of revolution:

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{\pi / 2}\left(25 \cos ^{2} x-\cos ^{2} x\right) d x= \\
& =24 \pi \cdot \int_{0}^{\pi / 2} \cos ^{2} x d x= \\
& =12 \pi \cdot \int_{0}^{\pi / 2}(1+\cos 2 x) d x= \\
& =\left.12 \pi \cdot\left(x+\frac{\sin 2 x}{2}\right)\right|_{0} ^{\pi / 2}= \\
& =12 \pi \cdot\left(\frac{\pi}{2}+0\right)-(0+0)=6 \pi^{2}
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_21-4 »$ Categories: Kuznetsov's Problem Book Integrals Problem 21 | Integrals

Ukrainian Banner Network

- Last edited on this page: 04:11, 25 June 2010.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 21-5

## Material from PlusPi