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Find the perimeter of an astroid with equation x^(2/3) + y^(2/3) = 64. (Hint: find the arclength of the portion of the curve in the first quadrant and use symmetry).

Firstly, rewrite the equation of this astroid in parametric form,

ie we have x =512 (cos t)^3 , y = 512 (sin t)^3

The section of the curve in the first quadrant is fully described by t from 0 to pi/2 inclusive.

Hence, arc length of this particular section of the curve is given by

integrate {t=0 to pi/2} sqrt[ (dy/dt)^2 +(dx/dt)^2 ] dt

=integrate {t=0 to pi/2} sqrt[ (1536 sin ^2 t * cos t)^2 +(-1536 cos ^2 t * sin t)^2 ] dt

=1536* integrate {t=0 to pi/2} sqrt[ sin ^4 t * cos^2 t + cos ^4 t * sin^2 t ] dt

= 1536* integrate {t=0 to pi/2} sqrt[ (sin ^2 t * cos^2 t) (sin ^2 t + cos^2 t) ] dt

= 1536* integrate {t=0 to pi/2} sqrt[ (sin ^2 t * cos^2 t)] dt

= 1536* integrate {t=0 to pi/2} sint cost dt

= 1536* integrate {t=0 to pi/2} 1/2* sin(2t) dt

= 1536* integrate {t=0 to pi/2} 1/2* sin(2t) dt

= 768 * integrate {t=0 to pi/2} sin(2t) dt

= 768 * [-1/2 * cos(2t)] {t=0 to pi/2}

= 768 * [ 1/2 - (-1/2)] = 768 units

Hence, the total perimeter of the asteroid is given by 768 *4 =3072 units (shown)

Hope this helps. Peace.

Calculus

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