The difference between Work Energy Diagram and Integral of Force equaling Work

You have to pay careful attention to the definitions. Work is the integral of the force over the displacement, not the time. These are different. If you want to solve this by integrating the force, you'd have to calculate $$W = int_{x(0)}^{x(2)} F(x),mathrm{d}x$$ and not $int F(t),mathrm{d}t$. However, we can use a substitution to change one integral into the other: $$W = int_{x(0)}^{x(2)} F(x),mathrm{d}x = int_0^2 F(t),frac{mathrm{d}x}{mathrm{d}t} ,mathrm{d}t =int_0^2 F(t),v(t),mathrm{d}t$$ The rest should be no problem for you.