Find the area enclosed by the cardioid r = 1 − sin θ

Question

Find the area enclosed by the cardioid r = 1 − sin θ (4 points)

answer choices

	3π
	6π
	12π

Answer 1

The area element is the wedge dA =1/2r²dθ, so your integral is

integral from 0 to 2pi of (1-sin(θ))²dθ which leads to 3/4(θ) evaluated from 0 to 2pi

most of the integral cancels when you put in the bounds and you have to do the 1/2(1-cos(2θ)) sub for sin²(θ)

Good luck!

Answer 2

Area = 1/2∫r²dθ from α to β

Plotting the on polar coordinates, the enclosed area is for θ = 0 to 2π

r² = sin²θ - 2sinθ + 1 = (1-cos(2θ)/2 -2sinθ + θ

A(θ) = θ/2 - sin(2θ)/4 + 2cos(θ) + θ from 0 to 2π

= π - 0/4 + 2 + 2π - [0 - 0 + 2 + 0] = 3π