Find the area of the region that lies inside the first curve and outside the second curve. r = 19 sin(θ), r = 10 − sin(θ)

A graph of the functions is attached. The area you need to calculate is colored with red.

The area can be calculated by the formula:


In order to find the limits of integration you need to calculate the two points at which the circumferences intersect:



19sinθ = 10 - senθ
20senθ = 10
senθ = 1/2
θ₁ = π/6 
θ₂ = 5π/6

Therefore:
=

=

=

=

=

=

[45√3/2 + 5√3 + 100π/3] - [-45√3/2 - 5√3 + 20π/3] =

55√3 + 80π/3 =

≈179.04

Hence, the area of the region that lies inside the first curve and outside the second curve is around 179 square units.