## Problem Statement

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

$$
y=3 \sin x, y=\sin x, 0 \leq x \leq \pi
$$
## Solution

Since the $O x$ axis is the axis of rotation, the volume is found using the formula:

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

Integration limits:

$x_{1}=0 ; x_{2}=\pi$

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

Integrals 21-3

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-05.jpg?height=1022&width=1122&top_left_y=1057&top_left_x=841)

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{\pi}\left(9 \sin ^{2} x-\sin ^{2} x\right) d x= \\
& =8 \pi \cdot \int_{0}^{\pi} \sin ^{2} x d x= \\
& =4 \pi \cdot \int_{0}^{\pi}(1-\cos 2 x) d x= \\
& =\left.4 \pi \cdot\left(x-\frac{\sin 2 x}{2}\right)\right|_{0} ^{\pi}=
\end{aligned}
$$

$$
=4 \pi \cdot(\pi-0)-(0-0)=4 \pi^{2}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-3$ »

Categories: Kuznetsov's Problem Book Integrals Problem 21 | Integrals | Problems for Checking

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## Problem Kuznetsov Integrals 21-4

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