Area enclosed by a polar curve
Area = ½ ∫αβ r² dθ.
Area swept out by the radius vector between θ = α and θ = β.
Worked examples
Circle r = a: area = ½ ∫02π a² dθ = πa².
Cardioid r = 1+cosθ: area = (3π/2).
Always check the limits to ensure the area swept is correct.
Frequently asked questions
Lemniscate?
r² = a² cos 2θ. Figure-of-eight curve. Total area = a².
Why ½?
Comes from the area of an infinitesimal sector: ½ r dr dθ.