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: 
For example, consider some water flow in the space, in which the density $\rho(x,t)$ fluctuates in space and in time. You might be interested in how the mass inside some fixed volume $V$ changes over time. The mass is equal to
$$
M(t)=\int\_V{\rho(x,t)\mathrm{d}x},
$$
therefore the "mass flow rate", using the rule you mentioned, is equal to
$$
\frac{\mathrm{d}}{\mathrm{d}t}M(t)=\int\_V \partial\_t\rho(x,t)\mathrm{d}x.
$$  

Other examples include energy, probability, momentum or state densities instead of mass density.
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