## Problem Statement

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

$$
y=x^{3}, y=x
$$
## Solution

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

Integrals 21-29

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

Express $x$ in terms of $y$ and find the limits of integration:

$$
\begin{aligned}
& y=x^{3} \Rightarrow x=\sqrt[3]{y} \\
& y=x \Rightarrow x=y \\
& y=x^{3}=x \Rightarrow x_{1}=0 ; x_{2}=1
\end{aligned}
$$

Find the volume of the solid as the difference in volumes of two solids of revolution:

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-57.jpg?height=1042&width=1011&top_left_y=1095&top_left_x=954)

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{1}(\sqrt[3]{y})^{2} d y-\pi \cdot \int_{0}^{1} y^{2} d y=\pi \cdot \int_{0}^{1}\left(y^{2 / 3}-y^{2}\right) d y= \\
& =\left.\pi \cdot\left(\frac{3}{5} \cdot y^{5 / 3}-\frac{y^{3}}{3}\right)\right|_{0} ^{1}=\pi \cdot\left(\frac{3}{5}-\frac{1}{3}\right)-(0-0)=\frac{4 \pi}{15}
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-29$ » Categories: Kuznetsov's Problem Book Integrals Problem 21 | Integrals | Problems for Checking

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Created by GeeTeatoo

## Problem Kuznetsov Integrals 21-30

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