## Task Condition

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

$$
y=-x^{2}+5 x-6, y=0
$$
## Solution

Since

the $O x$

integrals $21-1$

is

the

axis of rotation, the volume

is found

by

the formula:

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-01.jpg?height=1416&width=1405&top_left_y=1091&top_left_x=451)

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

Let's find the limits of integration:

$$
y=-x^{2}+5 x-6=0 \Rightarrow x_{1}=2 ; x_{2}=3
$$

Let's find the volume of the solid:

$$
\begin{aligned}
& V=\pi \cdot \int_{2}^{3}\left(-x^{2}+5 x-6\right)^{2} d x= \\
& =\pi \cdot \int_{2}^{3}\left(36-60 x+37 x^{2}-10 x^{3}+x^{4}\right) d x= \\
& =\left.\pi \cdot\left(36 x-\frac{60 x^{2}}{2}+\frac{37 x^{3}}{3}-\frac{10 x^{4}}{4}+\frac{x^{5}}{5}\right)\right|_{2} ^{3}= \\
& =\frac{\pi}{30}
\end{aligned}
$$

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

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

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## Problem Kuznetsov Integrals 21-2

## Material from PlusPi