## Task Condition

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

$$
y=\arccos x, y=\arcsin x, x=0
$$
## Solution

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

Integrals $21-30$

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

Let's find the limits of integration:

$$
\begin{aligned}
& y=\arccos x \Rightarrow x=\cos y \\
& y=\arcsin x \Rightarrow x=\sin y \\
& x=0 \Rightarrow y_{1}=\arcsin 0=0 \\
& \cos y=\sin y \Rightarrow y_{2}=\frac{\pi}{4} \\
& x=0 \Rightarrow y_{3}=\arccos 0=\frac{\pi}{2}
\end{aligned}
$$

We will find the volume of the solid as the sum of the volumes of two solids of revolution: $y=\arcsin x ; 0 \leq y \leq \frac{\pi}{4}$ and

$y=\arccos x ; \frac{\pi}{4} \leq y \leq \frac{\pi}{2}$

![](https://cdn.mathpix.com/cropped/2024_05_22_cfec74fe4e2360126fc5g-59.jpg?height=1416&width=939&top_left_y=1091&top_left_x=1027)

$$
V=\pi \cdot \int_{0}^{\pi / 4} \cos ^{2} y d y+\pi \cdot \int_{\pi / 4}^{\pi / 2} \sin ^{2} y d y=
$$

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+21-30$ » Categories: Kuznetsov's Problem Book Integrals Problem 21 | Integrals | Problems for Checking

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## Problem Kuznetsov Integrals 21-31

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