Find the length of cycloid from one cusp to the next , where `x=a(θ + sinθ) , y=a(1-cosθ)`

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#### Solution

Given curve :Cycloid `x=a(θ + sinθ) , y=a(1-cosθ)`

The length of given curve is :

`s=int_(θ1)^(θ2) sqrt((dx/(dθ))+(dy/(dθ))^2 dθ)`

`dx/(dθ)=a(1+cosθ) dy/(dθ)=a sin θ`

∴ `(dx/(dθ))^2 +(dy/(dθ))^2=a^2[1+2 cosθ +cos^2 θ+sin^2θ ]`

=`2a^2[1+cosθ]`

= `4a^2 [cos^2 θ/2]`

∴`sqrt((dx/dθ)^2+(dy/(dθ))^2)= 2a cos θ/2`

∴` s= int_-pi^pi 2acos θ/2 dθ`

= `2xxint_0^pi 2 a cos θ/2 dθ`

=` 4a [2 sin θ/2]_0^pi`

∴ `s= 8a`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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