$\"\"$

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Hypocycloid of four cusps is $\"\"$ .

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Graph:

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$\"\"$

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Consider $\"\"$ .

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Solve the equation for $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Consider $\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find the arclength of the hypocycloid.

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Definition of the arc length:If the curve $\"\"$ , $\"\"$ , then the length of the curve is defined as, $\"\"$ .

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Here $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Here integrand function has infinite discontinuity at $\"\"$ .

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$\"\"$

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$\"\"$ .

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Perimeter is four times the arclength by symmetry.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Perimeter of the hypocycloid is $\"\"$ units.