Question: The tensor of inertia of a solid sphere is $I\_{ii}=\frac{2}{5}MR^2$ about an axis passing through its CM. Why would the tensor of inertia of each hemisphere about that axis be $I\_{ii}=\frac{2}{5}mR^2$, where $m=\frac{M}{2}$ is the mass of the hemisphere? Is it because the tensor of inertia of each hemisphere would be expected to be $I\_{ii}=\frac{1}{5}MR^2$, but then $M = 2m$, hence $I\_{ii}=\frac{2}{5}mR^2$?

Answer: 
I guess a simple example would be the derivation of the continuity equation.

It will first be useful to define the phase space probability density, denoted by $f=f(q\_1,\dots,q\_{N},p\_1,\dots,p\_{N},t) \equiv f(\mathbf{x},t)$, that tells us the probability of finding a system near $(q\_1,\dots,q\_{N},p\_1,\dots,p\_{N})$ at a time $t$. Therefore:
\begin{equation}
\mathrm{d}N = f(\mathbf{x},t) \; \mathrm{d}x
\end{equation}
relates phase space volume elements to the number of particles $N$ in that volume. In other words, $f$ is the number of volume elements occupying the differential phase space volume $\mathrm{d}x$ and:
\begin{equation}
N = \int \mathrm{d}x \; f(\mathbf{x},t) 
\end{equation}
gives the number of particles in the ensemble.

Now, remember that the flux through a surface is given by:
\begin{equation}
j = \int f \mathbf{v} \cdot \mathrm{d} \mathbf{S}
\end{equation}
where $\mathbf{v}$ denotes the flow velocity of the particles. Subsequently, using the $N$-dimensional divergence theorem, the total flux out of a volume $V$ bounded by $\partial V$ is given by:
\begin{equation}
\oint\_{\partial V} f \mathbf{v} \cdot \mathrm{d} \mathbf{S} = \int\_V \mathbf{\nabla}\_{\mathbf{x}} \cdot \left( f \mathbf{v} \right) \; \mathrm{d}x
\end{equation}
where $\mathbf{\nabla}\_{\mathbf{x}}$ is the $2N$ dimensional gradient on the phase space:
\begin{equation}
\mathbf{\nabla}\_{\mathbf{x}} = \left(\frac{\partial}{\partial q\_1}, \dots , \frac{\partial}{\partial q\_{N}},\frac{\partial}{\partial p\_1}, \dots , \frac{\partial}{\partial p\_{N}} \right)
\end{equation}
On the other hand, using the rule of differentiation under the integral sign, we can write the rate of decrease of particles in the volume as:
\begin{equation}
\frac{d N}{d t} = - \frac{d}{d t} \int\_V \mathrm{d}x \; f = - \int\_V \mathrm{d}x \; \frac{\partial f}{\partial t}
\end{equation}
Therefore, demanding equality between the fourth equation and the sixth equation gives (because the flux out of the volume must equal the reate of decrease of particles in the volume):
\begin{equation}
\int\_V \mathbf{\nabla}\_{\mathbf{x}} \cdot \left( f \mathbf{v} \right) \; \mathrm{d}x = - \int\_V \mathrm{d}x \; \frac{\partial f}{\partial t}
\end{equation}
which is the continuity equation.

We also need to use the "differentiation under the integral sign" if we want to derive probability current in quantum mechanics. However, I have not taken this example, because I do not know your background and if/how much knowledge of QM you have. Also for Ehrenfest theorem in QM we need to use this rule.

Another (more mathematical) example when this is useful is for contour integration (and in particular for Cauchy's integral formula).
Possibly the most ubiquitous use is to calculate the derivative of a functional which is defined as an integral, e.g., of the Lagrangian along a path. Typical application---computing the equations of motion as the euler equations of a problem of the calculus of variations