For the polar curve r = 1 + cos θ, the area enclosed by the curve as θ runs from 0 to 2π is what value?

• A. π
• B. 2π
• C. 3π/4
• D. 3π/2

Answer: (D)

Area in polar coordinates is found with A = (1/2) ∫ r^2 dθ. Here, r = 1 + cos θ, and as θ goes from 0 to 2π the curve traces the entire cardioid, so we integrate over that full range.

Compute A = (1/2) ∫_0^{2π} (1 + cos θ)^2 dθ = (1/2) ∫_0^{2π} [1 + 2 cos θ + cos^2 θ] dθ.

Use cos^2 θ = (1 + cos 2θ)/2 to simplify:

The integrand becomes (3/2) + 2 cos θ + (1/2) cos 2θ.

Integrate term by term:

∫_0^{2π} (3/2) dθ = 3π,

∫_0^{2π} 2 cos θ dθ = 0,

∫_0^{2π} (1/2) cos 2θ dθ = 0.

So ∫_0^{2π} r^2 dθ = 3π, and the area is A = (1/2) · 3π = 3π/2.