area = (1/2)int[r^2*dθ] (0 < θ < 2π)
area = (1/2)int[(1 + cos(θ))^2*dθ]
area = (1/2)int[(1 + 2cos(θ) + cos^2(θ))*dθ]
area = (1/2)int[(1 + 2cos(θ) + [1/2 + (1/2)cos(2θ)])*dθ]
area = (1/2)int[(3/2 + 2cos(θ) + (1/2)cos(2θ))*dθ]
integrating:
area = (1/2)[3θ/2 + 2sin(θ) + (1/4)sin(2θ)]
evaluating for 0 < θ < 2π:
area = 3π/2
A1 = pi * r1^2 = pi * (3*sin(t))^2 = pi * [ 9 * (sin(t))^2 ] A2 = pi * r2^2 = pi * (1+sin(t))^2 = pi * [ 1 + 2*sin(t) + (sin(t))^2 ] A = A1 - A2 = pi * [ 8*(sin(t))^2 + 2*sin(t) +1 ] where: t = theta pi = 3.1415...
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Verified answer
area = (1/2)int[r^2*dθ] (0 < θ < 2π)
area = (1/2)int[(1 + cos(θ))^2*dθ]
area = (1/2)int[(1 + 2cos(θ) + cos^2(θ))*dθ]
area = (1/2)int[(1 + 2cos(θ) + [1/2 + (1/2)cos(2θ)])*dθ]
area = (1/2)int[(3/2 + 2cos(θ) + (1/2)cos(2θ))*dθ]
integrating:
area = (1/2)[3θ/2 + 2sin(θ) + (1/4)sin(2θ)]
evaluating for 0 < θ < 2π:
area = 3π/2
A1 = pi * r1^2 = pi * (3*sin(t))^2 = pi * [ 9 * (sin(t))^2 ] A2 = pi * r2^2 = pi * (1+sin(t))^2 = pi * [ 1 + 2*sin(t) + (sin(t))^2 ] A = A1 - A2 = pi * [ 8*(sin(t))^2 + 2*sin(t) +1 ] where: t = theta pi = 3.1415...