This question has already an accepted answer by sathak: however, I would like to share with other members what I have been able to find by looking in my textbooks and monographs on electrical noise. So, following the questions posed by the OP, here are my answers.
Why is it necessary to integrate over the channel to get the thermal noise? Wouldn't dividing the drain-source voltage by the drain current be enough to get the resistance used for thermal noise calculations? Or the differential resistance?
It is necessary to integrate the noise contribution of each single section of the channel since each of them must be weighted in a different manner, depending on its position along the channel. This is due to the fact that Field Effect Transistor (be it Metal Oxide Semiconductor or Junction) is an active device, therefore it provides amplification for every single electrical signal within its active region, no matter how small it is.
Precisely, let's consider the Johnson-Nyquist thermal noise generated inside each of its "infinitesimal" channel sections:
$$
\mathrm{d}v^2(\ell)=4k_\mathrm{B}T\Delta f\mathrm{d}R(\ell)=4kT\Delta f\frac{\mathrm{d}V(\ell)}{I_D}\label{1}\tag{n}
$$
where
- \$k_\mathrm{B}\$, \$T\$ and \$\Delta f\$ are respectively Boltzmann's constant, the absolute temperature of the channel of the device (assumed from here on to be a n-channel MOSFET) and the signal bandwidth considered, as is customary to do.
- \$\ell\$ is the coordinate of a point taken along the axis of the channel of the device (the geometry of the device is obviously assumed to be simple, for example rectangular in 2d or cylindrical in 3d): \$L\$ is assumed to be the entire channel length.
- \$V(\ell)\$ is the electric potential of the point \$\ell\$
- \$I_D\$ is the (stationary) drain current flowing though the device.
If we want to find the contribution of the noise \ref{1}, generated by every single \$\ell\$ channel section, to the total noise at the drain of the device we should perform circuit analysis.
Modeling \ref{1} as an ideal signal generator, we see that this generator has one terminal connected to a common gate MOSFET amplifier, which models the part of the global MOSFET comprised between the \$\ell\$ position and the drain (\$\ell=L\$) and whose characteristics depend on its channel length \$L-\ell\$. The other terminal is connected to the output conductance of another MOSFET, which models part of the global MOSFET comprised between the \$\ell\$ position and the source (\$\ell=0\$) and whose channel length is \$\ell\$. Thus the thermal noise generated inside each region of the device appears at the drain terminal amplified by a gain which is a function of \$\ell\$ and this is why it is necessary to integrate all the noise contributions of each infinitesimal resistive section of the channel. The following schematic could give some insight:
simulate this circuit – Schematic created using CircuitLab
The first to have used this fact in the noise analysis of the JFET transistor seem to have been Aldert Van Der Ziel in [1] (p. 1809): following the development for MOSFET device as offered by Cobbold in [1], §9.3.1. pp. 337-339, we have the following formula
$$
\overline{i_d^2}=-4k_\mathrm{B}T \Delta f \frac{\mu^\ast_n}{L^2}Q_n\label{2}\tag{N}
$$
where
- \$\mu^\ast_n\$ is the effective channel surface electron mobility,
- \$Q_n=\int_0^LW(\ell)Q_n(\ell)\mathrm{d}\ell\$ is the total channel inversion charge, obtained by integrating the channel inversion charge density \$Q_n(\ell)\$ along the channel length with given metallurgical channel width \$W(\ell)\$.
Is it correct that shot noise is a larger contribution than thermal noise for \$V_\mathrm{DS} > 50 \mathrm{mV}\$?
To (partially) answer to this question I must precise that, whenever a current of granular material flows trough the given section of a conductor (or of a tube, a chute and so on) a form of shot noise is nevertheless present. This is due to the fact that this kind of noise is due to the granularity of the flowing media (in this case electrons) and not on the physical properties of the medium through which it flows: in other word, shot noise originates from the statistical structure of the motion of granular materials. In the case of charges, a flows happens between two regions only if there is a difference between potentials (and considering that there are not perfect insulators, the condition could be assumed as necessary and sufficient): this is the basis of the assertion that shot noise appears across potential barriers. This implies that
there is shot noise even when \$V_\mathrm{DS}< V_\mathrm{GS} - V_p\$ and its value is always
$$
\overline{i^2_\mathrm{Sh}}=2qI_D\Delta f\label{3}\tag{Nsh}
$$
with the customary meaning of all symbols.
Now, for a MOSFET we have (Cobbold [1], §7.1.1 pp. 241-242 and §9.3.1 p. 338)
$$
I_D=-\frac{\mu^\ast_n}{L}\int\limits_0^LW(\ell)Q_n(\ell)\frac{\mathrm{d}V(\ell)}{\mathrm{d}\ell}\mathrm{d}\ell\label{4}\tag{ID}
$$
and therefore using equation \ref{2}, \ref{3} and \ref{4} we get
$$
\frac{\overline{i_d^2}}{\overline{i^2_\mathrm{Sh}}}=\frac{2k_\mathrm{B}T}{q}\frac{Q_n}{L \int\limits_0^LW(\ell)Q_n(\ell)\frac{\mathrm{d}V(\ell)}{\mathrm{d}\ell}\mathrm{d}\ell}\label{5}\tag{Noise comparation}
$$
As we can see, the limit value
$$
\frac{2k_\mathrm{B}T}{q} = 50\mathrm{mV}
$$
value appear in the equation for \ref{5}, and multiplies a term whose magnitude is a function of \$V_\mathrm{DS}\$: due to lack of time, I have not been able to estimate it, but it seems plausible that the limit stated by the OP for the dominance of the thermal channel noise on the shot noise can be formally justified.
[1] Richard S. C. Cobbold (1970), Theory and Applications of Field Effect Transistors, New York-London-Sydney-Toronto: John Wiley & Sons, pp. xv+534.
[2] Aldert Van Der Ziel (August 1962), "Thermal Noise in Field-Effect Transistors", in Proceedings of the IRE, vol. 50, no. 8, pp. 1808-1812, doi: 10.1109/JRPROC.1962.288221.
