So after the channel is pinched off in a MOSFET the equation that relates current and voltage through the MOSFET changes. What I am confused about though, is why in the derivation of this new formula the voltage is integrated from V = 0v to V = Vgs - Vt where Vgs is the potential difference between the source and the gate and Vt is the threshold voltage (at least that's what I think that's what those represent).

Anyway here is an visual representation of the circuit mentioned:

Here is also the integral I mentioned (equation 6.16):

Actually the integrals for the linear region and saturation region are the same. The result for saturation is just obtained as a special case with $V_{ds} = V_{gs} - V_T$.
The reason why $V_{gs} - V_T$ is used is simply because that is the maximum voltage that can drop along the channel. Put differently, this is the maximum possible $V_{ds}$ for the linear region which marks the transition to the saturation region. As a first approximation the current in the saturation region is constant, so the current at this boundary is exactly the current for saturation.
The same result is obtained, if the expression for the linear region is used with $V_{ds}$ being equal to $V_{gs} - V_T$.