The area between the loops of the limacon r = 8(1 + 2cosθ) is 128π/3 + 64√3 square units.
The polar curve r = 8(1 + 2cosθ) has two loops, one large and one small. The small loop is centered at (4,0) and the large loop is centered at (-4,0).
The equation of the curve :
r = 8 + 16cosθ
r = 8 + 16cosθ
0 = 8 + 16cosθ
cosθ = -1/2
θ = 2π/3 or 4π/3
We then set up the integral to find the area between the loops:
A = 1/2 ∫θ=2π/3 to 4π/3 [r(θ)]² dθ
A = 1/2 ∫θ=2π/3 to 4π/3 [8 + 16cosθ]² dθ
We simplify this integral by expanding the square and using trigonometric identities.
A = 128π/3 + 64√3
In conclusion, we can say that the area between the loops of the limacon r = 8(1 + 2cosθ) is 128π/3 + 64√3 square units.
complete question:
Find the area between the loops of the limacon r=8(1+2cosθ) r = 8 ( 1 + 2 cos θ ) .