Find the area of the region R bounded by the curve r = 6cos 0

Question

Find the area of the region R bounded by the curve r = 6cos 0

Answer 1

There are two ways to do this.

 

Method 1: Integral in polar coordinates:

 

The whole curve is traced once as θ goes from  0 to π

 

A = ∫_a^b (r²/2) dθ

 

= ∫₀^π 18 cos² θ dθ

 

= ∫₀^π (9 + 9 cos 2θ) dθ

 

= [9θ + (9/2) sin 2θ] |₀^π

 

= 9π

 

Method 2: To Cartesian Coordinates

 

Using x = r cos θ and y = r sin θ and x² + y² = r² we get

 

r = 6 cos θ

 

r² = 6r cos θ

 

x² + y² = 6x

 

x² - 6x + 9 + y² = 9

 

(x - 3)² + y² = 3²

 

So this is a circle of radius 3 with center at (3,0).

 

The area is thus A = πr² = 9π