Wednesday, March 30, 2022

March Calculus Madness Sweet Sixteen - Day 15: r = α * (1 - cos Θ)

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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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