Determine the area of the inner loop of the polar curve r = 1 − 2 sin (θ)

Question

Determine the area of the inner loop of the polar curve r = 1 − 2 sin (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 = 1 - 2sinθ; r < 0; 1 - 2sinθ < 0; sinθ > 1/2; π/6 < θ <5π/6

A = 1/2∫_π/6^5π/61 - 2sinθ)²dθ = 1/2∫_π/6^5π/6(1 - 4sinθ + 4sin²θ)dθ = 1/2∫_π/6^5π/6(3 - 4sinθ - 2cos2θ)dθ =

= 1/2(3θ + 4cosθ - sin2θ)_π/6^5π/6 = 1/2[(5π/2 + 4cos5π/6 - sin5π/3) - (3π/6 + 4cosπ/6 - sinπ/3) =

1/2(2π - 2√3 + √3/2 - 2√3 + √3/2) = π + 3√3/2 = 5.74