## Problem Statement

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

$$
y=x^{2}, y^{2}-x=0
$$
## Solution

Since the axis

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

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-22.jpg?height=1374&width=1448&top_left_y=1006&top_left_x=495)

Let's find the limits of integration:

$$
y=x^{2}=\sqrt{x} \Rightarrow x_{1}=0 ; x_{2}=1
$$

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

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{1}(\sqrt{x})^{2}-\left(x^{2}\right)^{2} d x= \\
& =\pi \cdot \int_{0}^{1}\left(x-x^{4}\right) d x= \\
& =\left.\pi \cdot\left(\frac{x^{2}}{2}-\frac{x^{5}}{5}\right)\right|_{0} ^{1}= \\
& =\pi \cdot\left(\frac{1}{2}-\frac{1}{5}\right)=\frac{3 \pi}{10}
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-11$ »

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

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## Problem Kuznetsov Integrals 21-12

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