How do you find the area of the region that lies inside the curves #r= 1+cos(theta)# and #r= 1-cos(theta)#?

1 Answer
Feb 26, 2015

First let us "see" our area:

enter image source here

Basically you want the area of the two loops enclosed by the two curves (vertically along the vertical axis).

In general the area in polar form is:
#1/2int_(theta_1)^(theta_2)r^2d(theta)# (have a look to any maths book on calculus/analytical geometry).

In this case let us start with the upper loop; you have:

#0->pi/2# for the red line + #pi/2->pi# for the blue line
area red shaded + area blue shaded

enter image source here
Area1= #1/2int_0^(pi/2)(1-cos(theta))^2d(theta)+1/2int_(pi/2)^(pi)(1+cos(theta))^2d(theta)#
So you get:
enter image source here

Hope it helps (check my maths)