Given the force field $\mathrm{F}$, find the work required to move an object on the given orientated curve.
$F=\langle y, x\rangle$ on the parabola $y=3 x^{2}$ from $(0,0)$ to $(2,12)$
The amount of work required is
(Simplify your answer.)

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 \(\boxed{-\frac{asinh(12)}{192} + \frac{289\sqrt{145}}{16}}\).