## Task Condition

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

$$
y=\sin \frac{\pi x}{2}, y=x^{2}
$$
## Solution

Since

$O x$

is

the axis of rotation, the volume is found using the formula:

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-32.jpg?height=1334&width=1514&top_left_y=1166&top_left_x=405)

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

Let's find the limits of integration:

$$
y=x^{2} \sin \left(\frac{\pi x}{2}\right) \Rightarrow x_{1}=0 ; x_{2}=1
$$

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

$$
\begin{aligned}
& V=\pi \int_{0}^{1} \sin ^{2}\left(\frac{\pi x}{2}\right) d x-\pi \int_{0}^{1}\left(x^{2}\right)^{2} d x= \\
& =\pi \int_{0}^{1}\left[\sin ^{2}\left(\frac{\pi x}{2}\right)-x^{4}\right] d x= \\
& =\pi \int_{0}^{1}\left[\frac{1-\cos (\pi x)}{2}-x^{4}\right] d x= \\
& =\left.\pi\left(\frac{x}{2}-\frac{\sin (\pi x)}{2 \pi}-\frac{x^{5}}{2}\right)\right|_{0} ^{1}=0.3 \pi
\end{aligned}
$$

Answer: \( V=0.3 \pi \)

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� $\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{\%} 0 \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD}$ $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-16 \%$ Categories: Kuznetsov's Problem Book Integrals Problem 21 | Integrals

- Last edited: 04:58, June 11, 2010.
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## Problem Kuznetsov Integrals 21-17

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