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Hypocycloid arc length integral for three rotations of the smaller circle, parametric integral.
We set up the parametric arc length integral for the three pointed hypocycloid given by a smaller circle rolling within a larger circle, with the large radius R=3 and the small radius r=1. To set the integral up, we plug the equations of the hypocycloid into the parametric arc length formula, which requires differentiating each coordinate equation, then the integrand is the square root of the sum of the squares of the derivatives. For an integral involving a square root with a sum of squared derivatives inside, our chances are usually poor to compute the integral analytically, but it turns out this one can be done in exact form!
We expand and simplify inside the square root, applying the pythagorean identity (sin(t))^2+(cos(t))^2=1 twice, then we recognize the identity for the cosine of the sum of two angles cos(x+y)=cos(x)cos(y)-sin(x)sin(y) and apply the identity to simplify the integral to the integral of sqrt(1-cos(3t)).
This integral is non-trivial, and we approach the integral by multiplying by the conjugate in the numerator and denominator to take advantage of the same pythagorean identity one more time. Then we end up with the square root of the square of the sine function in the numerator, with sqrt(1+cos(3t)) in the denominator.
Unfortunately, the square root of the square of the sine function gives us the absolute value of sine in the integral, so we have to investigate the integration interval and split up the integral into two pieces: one for which the sine is positive and one for which the sine is negative.
Once this is done, we modify constants for using the chain rule backwards, guess the antiderivatives and evaluate across the limits of integration to obtain the arc length of a hypocycloid.
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