Find the length of the curve from 0 to 1

[jorgegalvan93]

Homework Statement

find the length of the curve… r(t) = <4t, t^(2) + 1/6(t)^(3)> from 0≤t≤1

Homework Equations

L(t) = ∫a to b √(dx/dt)^(2) +(dy/dt)^(2) + (dz/dt)^(2))dt

The Attempt at a Solution

After taking the derivative of all components of the curve and finding the magnitude…
∫ 0 to 1 √(16+4t^(2) + 1/324(t)^(4))dt

And I can't manage to simplify any further.
I tried, doing ∫ 0 to 1 √(t)^(4) + 1296t^(2) + 5184

I don't know how to go any further in simplifying and pulling some t's out of the radical.
I know, it's some algebra, but I have no idea how to go about this from here.

 

[haruspex]

∫ 0 to 1 √(16+4t^(2) + 1/324(t)^(4))dt

The seem to be a few errors in the (dy/dt)² term.
OTOH, corecting them makes the integrand look even worse.