Tuesday, March 29, 2022

March Calculus Madness Sweet Sixteen - Day 14: The Arc Length of a Spiral

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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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