How can we find the arc length of the curve?

Question

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$

Answer 1

Length of a curve from $t=a$ and $t=b$ is given by $$\int_a^b \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} dt$$ provided $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ exists and are continuous.

In your problem, $x(t) = \dfrac{t^3}3 - t$, $y(t) = t^2$, $a = 0$ and $b = 1$.