Question

Find the area enclosed by the curve \({r^2} = 9\cos 5\theta \).

Short answer

The area enclosed by the curve is \(18\) square units.

Step-by-step solution

Definition of the parametric equation

A parametric equation in mathematics specifies a set of numbers as functions of one or more independent variables known as parameters.

Sketch the graph for given equation

Graph for \({r^2} = 9\cos 5\theta \)is shown below:

Find the area of petal

One petal lying between\( - \pi /10 \le \theta \le \pi /10\).

Then the area of one petal will be \({A_1} = \int_{ - \pi /10}^{\pi /10} {\frac{{{r^2}}}{2}{\rm{ }}} d\theta \).

As there are total 10 petals, so the total area will be \(Area = 10\int_{ - \pi /10}^{\pi /10} {\frac{{{r^2}}}{2}{\rm{ }}} d\theta \).

Find the area by using\({r^2} = 9\cos 5\theta \).

\(\begin{aligned}{c}Area = 10\int_{ - \pi /10}^{\pi /10} {\frac{{9\cos 5\theta }}{2}} {\rm{ }}d\theta \\ = 10\left( {\frac{{9\sin 5\theta }}{{10}}} \right)_{ - \pi /10}^{\pi /10}\\ = 10\left( {\frac{9}{{10}} \times 2} \right)\\ = 18\end{aligned}\)

Therefore, the required area of the curve is \(18\) square units.