Determine the area of the inner loop of the polar curve r = 2 + 4 cos (θ).

Question

Determine the area of the inner loop of the polar curve r = 2 + 4 cos (θ).

Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482)

Answer 1

r = 0 when cos(θ) = -1/2, which occurs at θ = 2π/3 and 4π/3

We use I[a, b] for the integral from a to b and E[a, b] for the evaluation from a to b

Then, as r dr dθ = dydx, we integrate

I[2π/3, 4π/3]I[0,2+4 cos(θ)] r dr dθ =

I[2π/3, 4π/3] 1/2r²E[0,2+4 cos(θ)] dθ =

I[2π/3, 4π/3] 2 + 8 cos(θ) + 8 cos²(θ) dθ

As cos(2θ) = 2 cos²(θ) - 1, 8 cos²(θ) = 4 cos(2θ) + 4

Then, I[2π/3, 4π/3] 2 + 8 cos(θ) + 8 cos²(θ) dθ =

I[2π/3, 4π/3] 6 + 8 cos(θ) + 4 cos(2θ) dθ =

6θ + 8 sin(θ) + 2 sin(2θ) E[2π/3, 4π/3] =

4π - 8√3 + 2√3 =

4π - 6√3 =

2.174