Why does the arc length formula not work for this parametric curve?

Question

I am trying to find the arc length for the parametric equations $x=\cos^3t,\,y=\sin^3t$, for $t\in[0,\,2\pi]$.

This interval of $t$-values traces the curve exactly once, yet if you use the standard formula for the arc length of a parametric curve ($\int_0^{2\pi}\sqrt{\dot{x}^2+\dot{y}^2}dt$), it gives you an answer of $0$ which is obviously wrong.

If, instead of doing this, I simply multiplied the arc length between $t=0$ and $t=\pi/2$ by $4$, I would get the correct answer ($6$). Can someone explain why the normal formula does not work in this instance?

Answer 1

if you use the standard formula for the arc length of a parametric curve ($\int_0^{2\pi}\sqrt{\dot{x}^2+\dot{y}^2}dt$), it gives you an answer of $0$ which is obviously wrong.

The integrand is always non-negative, so the integral can only be zero if the integrand is zero, which is not the case.

We can only guess what went wrong, presumably integrating over a singularity or picking the wrong branch of the square root. Like you used $\sqrt{t^2}=t$ which is not correct if $t< 0$. What's correct is $\sqrt{t^2} = |t|$.

$|t|$ as part of the integrand can be treated by handling different cases of $t\leqslant 0$ and $t\geqslant 0$. This means you can split the integral accordingly, so that $|t|$ can be replaced by either $-t$ or $+t$.