Matching Areas

Take a point \(P\) on the graph of \(y=2x^2,\) and draw a line going through \(P\) parallel to the \(y\)-axis. Then call the red area \(A:\) the area bounded by this line, the graph of \(y=x^2,\) and the graph of \(y=2x^2.\)

Now, draw a line going through \(P\) parallel to the \(x\)-axis. Then call the blue area \(B:\) the area bounded by this line, the graph of \(y=2x^2,\) and the graph of \(y=f(x).\)

This function \(f(x)\) is continuous and displays the unique property that for every point \(P\) on \(y = 2x^2,\) the two areas \(A\) and \(B\) are equal.

Find \(f(x)\) and evaluate \(f(12).\)