Can anyone help me with this question?

Question

Sketch the graph of the limacon r = 3 – 2 cosθ and the circle r = 2. Shade the region that is inside of the

limacon and outside of the circle. Find the area of the shaded region.

Answer 1

Use a graphic calculator. Make sure that you are in polar coordinates mode....

Assuming that you can find algebraically the points of intersection

Area = ∫_7π/6^11π/2 ∫₄^3-2sinθ r d r d θ

= ∫_7π/6^11π/2 { [ (1/2) r² |₄^3-2sinθ ] } d θ

=(1/2) ∫_7π/6^11π/2 [ (3 -2sinθ )² -16 ] d θ

= (1/2) ∫_7π/6^11π/2 [ -5 - 12 sin θ -2cos2θ ] d θ

= ( 39√3 −10π) / 6