An astroid is the portion of the curve defined by x=a ^ 3

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

An astroid is the portion of the curve defined by x=acos⁡3θ,y=asin⁡3θ(0≤θ≤π2),x=a\cos^{3}\theta,\qquad y=a\sin^{3}\theta\qquad (0\le \theta\le \tfrac{\pi}{2}), where a>0a>0 is a constant. Address the tasks below to determine the exact length of this quarter–astroid. (a) Compute dxdθ\dfrac{dx}{d\theta} and dydθ\dfrac{dy}{d\theta}. (b) Show that (dxdθ)2+(dydθ)2=9a2sin⁡2θcos⁡2θ\left(\dfrac{dx}{d\theta}\right)^{2}+\left(\dfrac{dy}{d\theta}\right)^{2}=9a^{2}\sin^{2}\theta\cos^{2}\theta. (c) Use the result from part (b) to obtain a simplified integrand for the arc–length integral, and then evaluate the definite integral to find the exact length of the curve on 0≤θ≤π20\le\theta\le\tfrac{\pi}{2}. Express your final answer in terms of aa. Show all steps clearly.

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