Find the area of the region that is inside the graph of r = 1 + sin θ and outside of the graph of r = 1.

Question

Find the area of the region that is inside the graph of r = 1 + sin θ and outside of the graph of r = 1. (4 points)

answer choices

	1.393
	1.785
	2.785
	5.571

Answer 1

A_{1 -} A₂ = ∫ ^π ( 1/2 r₁ ^ ² —1/2 r ₂ ^{^ 2} ) dθ

⁰

⁼ ∫ ^π 1/2 [ ( 1 ^{^ 2} − ( 1 − sin θ ) ^{^ 2} ] dθ

⁰

⁼ - 1/2 ∫ ^π ( 2 sin θ + sin ^{^ 2} θ ) dθ

⁰

^θ= - 1/2 [ - 2 cos θ − 1/2 θ − 1/4 sin 2 θ ] | ^π

0

= - 1/2 [ - 2 (-1 ) + π / 2 - 1/4 ( 0 ) ] −

1/2 [ - 2 ( 1 ) + 1/2 ( 0 ) - 1/4 ( 0 ) ]

= 1/2 ( 2 + π / 2 ) - 1/2 ( - 2 )

= 1 + π / 4 + 1

= 2 + π / 4

= 2. 7853