Continuity equation including mass generated

Question

I don't really conceptually understand why you integrate the generated mass from $a$ to $b$. I understand that you have to take account that it's in within $\left[a,b\right]$, but not why you integrate it.

Answer 1

The equation $\delta m = r(x,t) A \, \delta x \, \delta t$ tells you that mass is being created on every part of the segment $[a,b]$, or at least on every part where $r(x,t) > 0$. That is, a little bit of mass is being created in the neighborhood of $x = a$, a little bit in the neighborhood of $x=b$, a little bit in the neighborhood of $x = a + (b - a)/3$, and so forth in every other neighborhood within $[a,b]$ simultaneously. To know how fast mass is being created on the whole segment $[a,b]$, you have to add up all those little bits. That is what the integral does.