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Welcome to March Calculus Madness!
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r = α * (1 - cos Θ)
r = α * (1 - cos Θ)
r^2 = α^2 * (1 - 2 * cos Θ + cos^2 Θ)
dr/dΘ = α * sin Θ
(dr/dΘ)^2 = α^2 * sin^2 Θ
Area from 0 ≤ Θ ≤ 2*π
1/2 * ∫ α^2 * (1 - cos Θ)^2 dΘ from Θ = 0 to Θ = 2*π
= α^2/2 * ∫ 1 - 2 * cos Θ + cos^2 Θ dΘ from Θ = 0 to Θ = 2*π
= α^2/2 * ∫ 1 - 2 * cos Θ + 1/2 * cos(2*Θ) + 1/2 dΘ from Θ = 0 to Θ = 2*π
= α^2/2 * (3/2 * Θ - 2 * sin Θ + 1/4 * sin(2*Θ) from Θ = 0 to Θ = 2*π)
= 3/2 * π * α^2
Arc Length from 0 ≤ Θ ≤ 2*π
r^2 + (dr/dΘ)^2
= α^2 * (1 - 2 * cos Θ + cos^2 Θ) + α^2 * sin^2 Θ
= α^2 - 2 * α^2 * cos Θ + α^2 * (cos^2 Θ + sin^2 Θ)
= α^2 - 2 * α^2 * cos Θ + α^2
= 2 * α^2 - 2 * α^2 * cos Θ
= 2 * α^2 *(1 - cos Θ)
Arc Length:
∫ 2 * α^2 *(1 - cos Θ) dΘ from Θ = 0 to Θ = 2*π
= α * √2 * ∫ √(1 cos Θ) dΘ from Θ = 0 to Θ = 2*π
= α * √2 * 4 * √2
= 8 * α
Eddie
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