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Welcome to March Calculus Madness!
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The Length of a Spiral from 0 ≤ Θ ≤ m
The equation of a spiral: r = α * Θ
The arc length of a polar equation r(Θ): ∫ √(r(Θ)^2 + (dr/dΘ)^2) dΘ
r = α * Θ
r^2 = α^2 * Θ^2
dr = α dΘ
(dr/dΘ)^2 = α^2
∫ √(α^2 * Θ^2 + α^2) dΘ from Θ = 0 to Θ = m
= α * ∫ √(Θ^2 + 1) dΘ from Θ = 0 to Θ = m
= α/2 * ( ln|Θ + √(1 + Θ^2)| + Θ * √(1 + Θ^2) for Θ = 0 to Θ = m)
(see below)
= α/2 * ( ln|m + √(1 + m^2)| + m * √(1 + m^2) )
Aside:
∫ √(1 + Θ^2) dΘ
Let Θ = tan x
dΘ = sec^2 x dx
∫ √(1 + Θ^2) dΘ
= ∫ √(1 + tan^2 x) * sec^2 x dx
= ∫ √(sec^2 x) * sec^2 x dx
= ∫ sec^3 x dx
= 1/2 * ∫ sec x dx + (sec x * tan x)/2 + C
(per reduction integration rule for sec x)
= 1/2 * ln|tan x + sec x| + 1/2 * sec x * tan x + C
with:
Θ = tan x
arctan Θ = x
sec(arctan Θ) = sec x
√(1 + x^2) = sec x
= 1/2 * ln|Θ + √(1 + Θ^2)| + 1/2 * Θ * √(1 + Θ^2) + C
Eddie
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