## Task Condition

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

$$
y=x^{2}, y=1, x=2
$$
## Solution

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

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

Let's find the limits of integration: From the problem statement, we already have: $x_{2}=2$ Now let's find the lower limit:

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

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

$$
\begin{aligned}
& V=\pi \cdot \int_{1}^{2}\left(x^{2}\right)^{2}-(1)^{2} d x= \\
& =\pi \cdot \int_{1}^{2}\left(x^{4}-1\right) d x= \\
& =\left.\pi \cdot\left(\frac{x^{5}}{5}-x\right)\right|_{1} ^{2}= \\
& =\pi \cdot\left(\frac{2^{5}}{5}-2-\frac{1^{5}}{5}+1\right)= \\
& =\pi \cdot\left(\frac{32}{5}-\frac{1}{5}-1\right)=\pi \cdot \frac{32-1-5}{5}=\frac{26 \pi}{5}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-28.jpg?height=1356&width=834&top_left_y=984&top_left_x=1111)

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-14$ »

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

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## Problem Kuznetsov Integrals 21-15

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