Anonymous
Sketch r=1+sin θ (a cardioid) and r=1 (a circle). They intersect when sinθ = 0, i.e. when θ=0 and θ=π.
From the sketch you should see that for an elementary sector in the required region, the radius runs from 1 to 1+sinθ. And θ runs from 0 to π.
A = ∫[θ=0 to θ=π]∫[from r=1 to r= 1+sinθ)] r dr dθ
= ∫[θ=0 to θ=π] r² |[from r=1 to r= 1+sinθ)] dθ
= ½∫[θ=0 to θ=π] ((1+sinθ)² - 1²) dθ
= ½∫[θ=0 to θ=π] (2sinθ + sin²θ) dθ
= ½(-2cosθ + ½θ - ¼sin(2θ)) |[θ=0 to θ=π]
= (1/8)(2θ - 8cosθ - sin(2θ)) |[θ=0 to θ=π]
= (1/8)(2π - 8cosπ - sin(2π)) - (2*0 - 8cos(0) – sin(2*0))
= (1/8)(2π - 8(-1) - 0) - (- 8*1 – 0))
= (1/8)(2π +8 + 8)
= (π+8)/4
But check my working of course.