A particle in the \(xy\)-plane moves from the origin \((0,0)\) to a final position \((x_0, y_0)\) along the parabolic path \(y = kx^2\), where \(k\) is a positive constant. During this displacement, the particle is acted upon by a two-dimensional force \(\vec{F} = ay\hat{i} + bx\hat{j}\), where \(a\) and \(b\) are constants. Which of the following expressions represents the work done by the force on the particle?