In polar coordinates the cardioid is r=a(1+cosθ).
The area of an infinitesimal sector of the cardioid is ½r2dθ=½a2(1+cosθ)2dθ.
The integral which gives us the area A is therefore the sum of these infinitesimals:
A=½a20∫2π(1+2cos(θ)+cos2(θ))dθ; cos2θ=½(cos(2θ)+1),
A=½a20∫2π(1+2cosθ+½(cos(2θ)+1))dθ=½a2[θ+2sinθ+¼sin(2θ)+θ/2]02π=3πa2/2.
This only involves a single integral. Generally A=∫dA=∫∫f(x,y)dxdy=∫∫f(r,θ)rdrdθ. Therefore:
A=θ=0∫θ=2πr=0∫r=a(1+cos(θ))rdrdθ=0∫2π[r2/2]0a(1+cos(θ)dθ=0∫2π(½a2(1+cos(θ))2)dθ.
A=½a20∫2π(1+2cos(θ)+cos2(θ))dθ=3πa2/2, which is what we had before.