Question
Question: How do you find the area of one petal of \(r=\cos 5\theta \)?...
How do you find the area of one petal of r=cos5θ?
Solution
We first need to consider the graph of the equation r=cos5θ. Then we have to put r=0 in the given equation to get cos5θ=0. The two consecutive solutions of this equation will give the θ coordinate of the endpoints of one petal, as θ1 and θ2. Then using the formula for the area in polar coordinates, which is given by A=21∫θ1θ2r2dθ we can determine the required area of one petal.
Complete step by step answer:
The equation given is
⇒r=cos5θ
As we can see that the above equation is a relation between the variables r and θ, which are the polar variables. The graph of the above equation is as shown in the below diagram.
We can clearly see that the graph of the equation r=cos5θ consists of five identical petals. Let us consider the horizontal petal and try to evaluate its area. For this, we need to determine the polar coordinates of the end points of the horizontal petal.
We can see that the petal starts from and ends at the origin. So the r coordinate for both of its end points is equal to 0. Therefore, putting r=0 in the given equation we get
⇒0=cos5θ⇒cos5θ=0
Now, we know that the solution of the equation cosx=0 is x=(2n+1)2π. So the solution of the above equation is given by
⇒5θ=(2n+1)2π⇒θ=(2n+1)10π
Since the horizontal petal lies in the first and the fourth quadrant, we get θ=−10π and θ=10π.
Now, we know that the area in terms of the polar coordinates is given by
⇒A=21∫θ1θ2r2dθ
Substituting r=cos5θ, θ1=−10π and θ2=10π, we get the area of the horizontal petal as
⇒A=21∫−10π10π(cos25θ)dθ
We know that 2cos2x=cos2x+1. Substituting x=5θ, we get