How would you find the length of the curve x = t^2 y = t^3 bounds are 0

Question

How would you find the length of the curve x = t^2  y = t^3 bounds are 0 < t < 2.

anonymous · Answer

The length of the curve is$$\int\limits_{0}^{2}|v(t)|dt$$

blockcolder · Answer

Since the curve has given parametric equations, the arc length formula becomes:
$$\int_{0}^{2} \sqrt{\left (\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$$

anonymous · Answer

first evaluate 't' froom the equation, then compare t, so u can find relation berween x and y like x=y^(2/3), then draw a graph, u can find a length of curve,

anonymous · Answer

$$
\int_0^2
   \sqrt{x'(t)^2+y'(t)^2} \,
   dt=\int_0^2\sqrt{9 t^4+4 t^2}dt=\
\int_0^2t \sqrt{9 t^2+4}dt=\
$$
It is easy now. Do the integral by substitution, you should get

$$\frac{8}{27} \left(10
   \sqrt{10}-1\right)

$$