Question

How do you find the area of the region bounded by the polar curve ${r^2} = 4\cos \left( {2\theta } \right)$.

Answer 1

In this question we will find the area bounded by the trigonometric equation. To find the area bounded by the curve, we will use double integration. We will integrate the given trigonometric function between the upper and lower limit of $a$ and $b$. We use the integration formula to find the area bounded by the curve in infinity small wedges. So we calculate the area of infinite numbers of rectangles between the limit $x = a$ and $x = b$, where $f\left( x \right)$ is equal to the height of each rectangle. Polar coordinate also follows the same rule. Only the main difference between the polar curve and normal curve is that, in the polar curve we find the area of the slice of the circle.
The area of region bounded by the polar curve is,
$\dfrac{1}{2}\int\limits_a^b {{r^2}} d\theta$
We will integrate the given trigonometric function between $x = - \dfrac{\pi }{4}$ to $x = \dfrac{\pi }{4}$ because the graph of the polar curve gets its cycle twice. If $\theta = \dfrac{\pi }{4}$ then $\cos \left( {2\theta } \right)$ will be like the graph of $\cos \left( {2\dfrac{\pi }{2}} \right) = \cos \dfrac{\pi }{2}$ . The radius of the circle will be zero. If we put our limit of integration from $x = - \dfrac{\pi }{4}$ to $x = \dfrac{\pi }{4}$ then we will find the area of the half curve. We will multiply the total area by 2 to find the whole area of the curve.

Complete step by step solution:
Step: 1 the given trigonometric equation of the polar curve is,
${r^2} = 4\cos \left( {2\theta } \right)$
We will find the area of each slice of the circle. We know that $r$ is the radius of each slice of the circle.
The area of the each slice bounded by the curve will be given as,
$A = \int\limits_{ - \dfrac{\pi }{4}}^{\dfrac{\pi }{4}} {4\cos \left( {2\theta } \right)} d\theta$
To find the area of the whole region bounded by the curve, we will multiply the area with two.
$\Rightarrow A = 2\int\limits_{ - \dfrac{\pi }{4}}^{\dfrac{\pi }{4}} {4\cos \left( {2\theta } \right)} d\theta$
Solve the integration to find the area of the curve.
$\Rightarrow A = 2\left( {\dfrac{1}{2}} \right)\int\limits_{ - \dfrac{\pi }{4}}^{\dfrac{\pi }{4}} {4\cos \left( {2\theta } \right)} 2d\theta$
Now solve the integration and substitute the limits to find the area.
$\Rightarrow A = 4\sin \left( {2\theta } \right){|^{\dfrac{\pi }{4}}}_{ - \dfrac{\pi }{4}}$
Solve the limit of the function to find the value of the area.
$\Rightarrow A = 4\sin \left( {2\dfrac{\pi }{4}} \right) - 2\sin \left( {2\left( { - \dfrac{\pi }{4}} \right)} \right) \\\ \Rightarrow A = 4\sin \left( {\dfrac{\pi }{2}} \right) - 4\sin \left( { - \dfrac{\pi }{2}} \right) \\\ \Rightarrow A = 4 - \left( { - 4} \right) \\\ \Rightarrow A = 8 \\\$

Final Answer:
Therefore the area of the region bounded by the polar curve is equal to 8.

Note:
Students are advised to first find the area of each slice of half of the circle and then multiply the area of half circle with 2 to find the total area of the circle. They must use the integration method to solve the problem. They should not make mistakes, while solving the limits of the equation.