Q. 33 - Calculus

Question

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one petal of $r=\cos 2\theta$

Step-by-Step Solution

Answer

The area bounded by one petal is $\frac{\mathrm{\pi }}{8} {\mathrm{unit}}^2$

Step 1: Given information

The curve is $r=\cos 2\theta$

Step 2: Area

The area bounded by the curve is,

$$\begin{gathered} A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{\frac{\mathrm{\pi }}{4}}{\left(\cos 2\theta \right)}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{\frac{\mathrm{\pi }}{4}}\left({\cos }^{2}2\theta \right)d\theta \\ A=\frac{1}{2}{\int }_0^{\frac{\mathrm{\pi }}{4}}\frac{\left(1+\cos 4\theta \right)}{2}d\theta \\ A=\frac{1}{2}{\left[\theta +\frac{\sin 4\theta }{4}\right]}_0^{\frac{\mathrm{\pi }}{4}} \\ A=\frac{1}{2}\left[\frac{\mathrm{\pi }}{4}+\frac{\sin 4{\displaystyle \frac{\mathrm{\pi }}{4}}}{4}-0\right] \\ A=\frac{\mathrm{\pi }}{8} {\mathrm{unit}}^2 \end{gathered}$$

The area is $A=\frac{\mathrm{\pi }}{8} {\mathrm{unit}}^2$