Find the area inside r=1+sin(theta) and outside r=1?

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.