## Task Condition

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

$$
r=\sin 3 \phi
$$
## Solution

![](https://cdn.mathpix.com/cropped/2024_05_22_17ecf51401f349d22271g-10.jpg?height=560&width=652&top_left_y=922&top_left_x=128)

We will use the formula for calculating the area of a region in polar coordinates:

$$
S=\frac{1}{2} \int_{\alpha}^{\beta} r^{2}(\phi) d \phi
$$

Clearly, to find the area of the entire figure, it is sufficient to calculate the area for \(0 \leq \phi \leq \frac{\pi}{3}\) and multiply it by 3.

$$
S=\frac{3}{2} \int_{0}^{\frac{\pi}{3}} \sin ^{2} 3 \phi d \phi=\frac{3}{2} \int_{0}^{\frac{\pi}{3}} \frac{1+\cos 6 \phi}{2} d \phi=\left.\left(\frac{3}{4} \phi+\frac{1}{8} \sin 6 \phi\right)\right|_{0} ^{\frac{\pi}{3}}=\frac{\pi}{4}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+16-6$ » Categories: Kuznetsov Problem Book Integrals Problem 16 | Integrals

Kuznetsov Problem Book Integrals 16-6 — PlusPi

Created by GeeTeatoo

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佂

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## Problem Kuznetsov Integrals 16-7

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