Which one or ones of the following integrals produces the area that lies inside the cardioid r = 4 + 4 cos θ and

Question

Which one or ones of the following integrals produces the area that lies inside the cardioid r = 4 + 4 cos θ and outside the circle r = 6? (5 points)

1. 
2. 
3. 

	I only
	II only
	II and III only
I got this answer	I and IIII only

Answer 1

The answer is 2 and 3.

2 is the obvious answer, as 4 + 4 cos θ > 6 when cos θ > 1/2 or -π/3 < θ < π/3

when we integrate r dr dθ, the change of variable from dy dx, with respect to r, we get 1/2r²

3 comes from the fact that the function in 2 is symmetric about θ = 0 as cos is even, and the integration is also symmetric about 0, so we can just double the integral over positive θ .

The first integral is the negative of the area outside the cardioid 4 + 4 cos θ and inside r = 6.