Step 1 :Find the intersection points of the two curves \(y=3x^2\) and \(y=4-x^2\). These points will give us the limits of integration for \(x\).
Step 2 :The intersection points are \((-1, 3)\) and \((1, 3)\).
Step 3 :Sort the \(x\)-values of the intersection points in ascending order to get the limits of integration for \(x\). The sorted \(x\)-values are \([-1, 1]\).
Step 4 :Integrate the function \(xy\) with respect to \(x\) first and then with respect to \(y\).
Step 5 :The double integral evaluates to zero.
Step 6 :Final Answer: \(\boxed{0}\)