## Problem Statement

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

$$
y=\arccos \frac{x}{5}, y=\arccos \frac{x}{3}, y=0
$$
## Solution

Since

the $O y$

axis

is

the axis

of rotation, the volume is found using the formula:

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-51.jpg?height=828&width=1514&top_left_y=1042&top_left_x=405)

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

Let's find the limits of integration:

$$
\begin{aligned}
& y=\arccos \frac{x}{5} \Rightarrow x=5 \cos y \\
& y=\arccos \frac{x}{3} \Rightarrow x=3 \cos y \\
& 5 \cos y=3 \cos y \Rightarrow y=\frac{\pi}{2}
\end{aligned}
$$

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

$$
V=\pi \cdot \int_{0}^{\pi / 2}(5 \cos y)^{2} d y-\pi \cdot \int_{0}^{\pi / 2}(3 \cos y)^{2} d y=
$$

$$
\begin{aligned}
& =(25-9) \pi \cdot \int_{0}^{\pi / 2} \cos ^{2} y d y= \\
& =16 \pi \cdot \int_{0}^{\pi / 2} \cos ^{2} y d y= \\
& =8 \pi \cdot \int_{0}^{\pi / 2}(1+\cos 2 y) d y= \\
& =\left.8 \pi \cdot\left(y+\frac{\sin 2 y}{2}\right)\right|_{0} ^{\pi / 2}=8 \pi \cdot\left(\frac{\pi}{2}+0\right)-(0+0)=4 \pi^{2}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� $\% \mathrm{D} 0 \% 9 \mathrm{~A} \% \mathrm{D} 1 \% 83 \% \mathrm{D} 0 \% \mathrm{~B} 7 \% \mathrm{D} 0 \% \mathrm{BD} \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 1 \% 86 \% \mathrm{D} 0 \% \mathrm{BE} \% \mathrm{D} 0 \% \mathrm{~B} 2 \mathrm{\%} 0 \mathrm{D} 0 \% 98 \% \mathrm{D} 0 \% \mathrm{BD}$ $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-26 \%$ Categories: Kuznetsov Problem Book Integrals Problem 21 | Integrals

- Last edited: 08:50, June 1, 2010.
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## Problem Kuznetsov Integrals $21-27$

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