## Task Condition

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

$$
x^{2}+(y-2)^{2}=1
$$
## Solution

$$
\begin{aligned}
& x^{2}+(y-2)^{2}=1 \\
& (y-2)^{2}=1-x^{2} \\
& y-2= \pm \sqrt{1-x^{2}} \\
& y= \pm \sqrt{1-x^{2}}+2
\end{aligned}
$$

Thus, we have two functions:

$$
y=\sqrt{1-x^{2}}+2
$$

and

$$
y=-\sqrt{1-x^{2}}+2=2-\sqrt{1-x^{2}}
$$

Since \( x^{2}+(y-2)^{2}=1 \) is the equation of a circle in the form \((x+a)^{2}+(y+b)^{2}=R\), its radius \( R \) is 1.

Therefore, the function is defined and integrable on \([-1 ; 1]\).

The volume of the solid formed by rotation is given by the formula:

$$
V=\pi \int_{a}^{b} f^{2}(x) d x \text { for the } O x \text { axis }
$$

We will compute the volume of the given figure as the difference of the volumes of two

Integrals 21-12

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-24.jpg?height=1205&width=728&top_left_y=971&top_left_x=1241)
functions:

$$
\begin{aligned}
& V=\pi \int_{-1}^{1}\left(\sqrt{1-x^{2}}+2\right)^{2} d x-\pi \int_{-1}^{1}\left(2-\sqrt{1-x^{2}}\right)^{2} d x= \\
& =\pi \int_{-1}^{1}\left(\left(1-x^{2}\right)+4 \sqrt{1-x^{2}}+4\right) d x-\pi \int_{-1}^{1}\left(4-4 \sqrt{1-x^{2}}+\left(1-x^{2}\right)\right) d x= \\
& =\pi\left(\int_{-1}^{1}\left(5-x^{2}+4 \sqrt{1-x^{2}}\right) d x-\int_{-1}^{1}\left(5-x^{2}-4 \sqrt{1-x^{2}}\right) d x\right)=
\end{aligned}
$$

$$
\begin{aligned}
& =\pi \int_{-1}^{1}\left(\left(5-x^{2}+4 \sqrt{1-x^{2}}\right)-\left(5-x^{2}-4 \sqrt{1-x^{2}}\right)\right) d x= \\
& =\pi \int_{-1}^{1} 8 \sqrt{1-x^{2}} d x=8 \pi \int_{-1}^{1} \sqrt{1-x^{2}} d x=\left.4 \pi\left(\arcsin x+x \sqrt{1-x^{2}}\right)\right|_{-1} ^{1}=2 \pi^{2}+2 \pi^{2}=4 \pi^{2}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\� $\% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{2} \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5$ $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-12$ "

Categories: Kuznetsov Integral Problems 21 | Integrals

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## Problem Kuznetsov Integrals 21-13

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