A particle moves in the xy-plane from the origin (0,0) to the point (x_0, y_0) along the parabolic path y = kx^2, where k, x_0, and y_0 are positive constants such that y_0 = kx_0^2.

During this displacement, the particle is acted upon by a two-dimensional force field given by \(\vec{F} = by\,\hat{i} + cx\,\hat{j}\), where b and c are non-zero constants with appropriate SI units.

Which of the following expressions correctly represents the integral for the total work done by this force on the particle as it moves along the specified path?