area of cardiode

Question

find by double integration the area of the cardioide  r=a(1+cos x)

Answer 1

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]₀^2π=3πa²/2.

This only involves a single integral. Generally A=∫dA=∫∫f(x,y)dxdy=∫∫f(r,θ)rdrdθ. Therefore:

A=_θ=0∫^θ=2π_r=0∫^{r=a(1+cos(θ))}rdrdθ=₀∫^2π[r²/2]₀^a(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.