Find the arc length of the astroid x(t)=cos^{3}t,\ y(t)=sin^{3}t

Question

Find the arc length of the astroid

x (t) = \cos^{3} t, y (t) = \sin^{3} t

.
Calculate the arc length of

\frac{1}{4}

of the astroid

(0 \leq t \leq \frac{π}{2})

Answer 1

$d s = \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}}$
$(\frac{d x}{d t}) = 3 \cos^{2} t (- \sin t)$
$(\frac{d x}{d t}) = 3 \sin^{2} t (\cos t)$
$d s = \sqrt{9 \cos^{4} t \sin^{2} t + 9 \sin^{4} t \cos^{2} t} d t$
$d s = 3 \sin t \cos t d t \sqrt{\cos^{2} t + \sin^{2} t} d t$
$d s = 3 \sin t \cos t d t$
$S = \int_{0}^{\frac{π}{2}} 3 \sin t \cos t d t = \frac{3}{2} {(\sin t)}^{2} ⌉_{0}^{\frac{π}{2}} = \frac{3}{2}$

Find the arc length of the astroid x(t)=cos^{3}t,\ y(t)=sin^{3}t

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