## Problem Statement

Calculate the areas of figures bounded by lines given in polar coordinates.

$$
r=\cos \phi, r=\sqrt{2} \sin \left(\phi-\frac{\pi}{4}\right),\left(-\frac{\pi}{4} \leq \phi \leq \frac{\pi}{2}\right)
$$
## Solution

We will construct the graphs of the given functions:

Find the points of intersection:

![](https://cdn.mathpix.com/cropped/2024_05_22_17ecf51401f349d22271g-17.jpg?height=1419&width=1290&top_left_y=987&top_left_x=657)

$$
\begin{aligned}
& \cos \varphi=\sqrt{2} \cos \left(\varphi-\frac{\pi}{4}\right) \\
& \cos \varphi=\cos \varphi+\sin \varphi \\
& \sin \varphi=0 \\
& \varphi=0
\end{aligned}
$$

From the graph, it is clear that the desired area is:

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+16-9$ »

Categories: Kuznetsov's Problem Book Integrals Problem 16 | Integrals

- Last edited: 13:30, 13 July 2010.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 16-10

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