## Task Condition

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

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

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

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

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

$$
y=x^{2} \Rightarrow x=\sqrt{y}
$$

From the problem statement, we already have: $y_{1}=0$.

Now find the upper limit:

$$
y=x^{2} ; x=2 \Rightarrow y_{2}=2^{2}=4
$$

Find the volume of the solid as the difference of the volumes of two solids of revolution:
Integrals 21-19

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-38.jpg?height=1345&width=948&top_left_y=1092&top_left_x=1002)

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{4}(2)^{2} d y-\pi \cdot \int_{0}^{4}(\sqrt{y})^{2} d y=\pi \cdot \int_{0}^{4}(4-y) d y= \\
& =\left.\pi \cdot\left(4 y-\frac{y^{2}}{2}\right)\right|_{0} ^{4}=\pi \cdot((16-8)-(0-0))=8 \pi
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-19$ »

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

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