Arc-length without bounds

Question

When I tried to calculate the Arc length of $r = (\cos^3x, \sin^2x)$, I used the arc length formula and got $\cos x \cdot \sin x(3\cos x+2)$. I do not get why we don't need the bounds (range) for the parameters in order to get the length. Could someone help? Thanks!

Answer 1

Hint

In parametric for $(x(t),y(t)$, the arc length is given by $$L=\int_{t_0}^{t_1} \sqrt{\left(\frac {dx}{dt} \right)^2+\left(\frac {dy}{dt} \right)^2}\,dt$$

So, as Doug M commented, the first problem is to compute the antiderivative $$f(t)=\int \sin (t)\cos(t)\sqrt{9\cos^2(t) + 4}\, dt$$ where you should notice that $\sin (t)\cos(t)$ is "almost" the derivative of $\cos^2(t)$ which reveals a quite obvious cange of variable.

When you will finish, then just apply to get $$L=|f(t_1)-f(t_0)|$$

By the way, the curve is entirely defined by $0 \leq t \leq \pi$.