Arc Length with Parametric Equations — Question 1

Question 1

Problem

Find the length of the astroid $x=\cos^3t$, $y=\sin^3t$.

Question 1 – Solution

Solution

1. The astroid is symmetric about both coordinate axes, so compute the length in the first quadrant, $0\le t\le\pi/2$, and multiply by four.

2. Differentiate: $$x'(t)=-3\cos^2t\sin t, \qquad y'(t)=3\sin^2t\cos t.$$

3. Simplify the speed: $$\begin{aligned} \sqrt{x'^2+y'^2} &=3\sqrt{\cos^4t\sin^2t+\sin^4t\cos^2t}\\ &=3\sqrt{\sin^2t\cos^2t(\cos^2t+\sin^2t)}\\ &=3|\sin t\cos t|. \end{aligned}$$ On $0\le t\le\pi/2$, both factors are nonnegative, so the speed is $3\sin t\cos t$.

4. The first-quadrant length is $$L_1=3\int_0^{\pi/2}\sin t\cos t\,dt =\frac32\left[\sin^2t\right]_0^{\pi/2} =\frac32.$$

5. Therefore, the total length is $$\boxed{L=4L_1=4\left(\frac32\right)=6}.$$