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.

Yes, two vertical lines can be complementary

Polar coordinates is 8,9