## Task Condition

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

$$
y=e^{1-x}, y=0, x=0, x=1
$$
## Solution

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

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

From the problem statement, we already have the limits of integration: $x_{1}=0 ; x_{2}=1$

Let's find the volume of the solid of revolution:

$$
\begin{aligned}
& V=\pi \cdot \int_{0}^{1}\left(e^{1-x}\right)^{2} d x= \\
& =\pi \cdot \int_{0}^{1}\left(e^{1} \cdot e^{-x}\right)^{2} d x= \\
& =\pi \cdot \int_{0}^{2} e^{2} \cdot e^{-2 x} d x= \\
& =\pi \cdot e^{2} \cdot \int_{0}^{1} e^{-2 x} d x= \\
& =\left.\pi \cdot e^{2} \cdot\left(-2 \cdot e^{-2 x}\right)\right|_{0} ^{1}= \\
& =2 \pi \cdot e^{2} \cdot\left(1-e^{-2}\right)=2 \pi \cdot\left(e^{2}-e^{2-2}\right)=2 \pi \cdot\left(e^{2}-1\right)
\end{aligned}
$$

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} \% \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}+21-10 »

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

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## Problem Kuznetsov Integrals 21-11

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