Step 1 :First, we parameterize the curve defined by the parabola \(y = 3x^2\). We let \(x = x\) and \(y = 3x^2\), so the parameterized curve is \(r = [x, 3x^2]\).
Step 2 :Next, we compute the derivative of \(r\) with respect to \(x\), which gives us \(dr = [1, 6x]\).
Step 3 :We then compute the magnitude of \(dr\), which is \(\sqrt{36x^2 + 1}\).
Step 4 :We also compute the force field \(F = [3x^2, x]\).
Step 5 :We then compute the dot product of \(F\) and \(dr\), which gives us \(9x^2\).
Step 6 :Finally, we compute the line integral of the force field along the curve, which is the work done by the force field. This is given by \(-\frac{asinh(12)}{192} + \frac{289\sqrt{145}}{16}\).
Step 7 :So, the amount of work required is \(\boxed{-\frac{asinh(12)}{192} + \frac{289\sqrt{145}}{16}}\).