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Find length of curve the r = sin^2(θ/2),0<=θ<=μ, a>0.

User Eden
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The length of a curve C is given by the integral,


\displaystyle\int_C\mathrm ds

where the line element ds is


\mathrm ds=\sqrt{\left((\mathrm dx)/(\mathrm dt)\right)^2+\left((\mathrm dy)/(\mathrm dt)\right)^2}\,\mathrm dt

where
x=x(t) and
y=y(t) are parameterizations of C.

In this case, we have


x(\theta)=r(\theta)\cos\theta


y(\theta)=r(\theta)\sin\theta

Differentiate with respect to
\theta to get


(\mathrm dx)/(\mathrm d\theta)=(\mathrm dr)/(\mathrm d\theta)\cos\theta-r(\theta)\sin\theta


(\mathrm dy)/(\mathrm d\theta)=(\mathrm dr)/(\mathrm d\theta)\sin\theta+r(\theta)\cos\theta


(\mathrm dr)/(\mathrm d\theta)=\sin\left(\frac\theta2\right)\cos\left(\frac\theta2\right)=\frac12\sin\theta

So the arc length is


\displaystyle\int_C\mathrm ds=\int_0^a√(\left(\frac12\sin\theta\cos\theta-\sin^2\left(\frac\theta2\right)\sin\theta\right)^2+\left(\frac12\sin\theta\sin\theta+\sin^2\left(\frac\theta2\right)\cos\theta\right)^2)\,\mathrm d\theta


=\displaystyle\int_0^a√(\sin^2\left(\frac\theta2\right))\,\mathrm d\theta


=\displaystyle\int_0^a\sqrt{\frac{1-\cos\theta}2}\,\mathrm d\theta

User Fentas
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