Question

A particle moving in the x direction is being acted upon by a net force f(x)=cx2, for some constant

Answer 1

The work-energy theorem states that the change in kinetic energy of the particle is equal to the work done on the particle:
$\Delta K = W$
The work done on the particle is the integral of the force on dx:
$W= \int\limits^{3L}_L {F(x)} \, dx = \int\limits^{3L}_L {cx^2} \, dx = \frac{26}{3}cL^3$
So, this corresponds to the change in kinetic energy of the particle.