What does this FTC scenario actually represent? Expounding on this question from my other thread
$$\frac{d}{dt}\int_1^2t^2dt$$
I want to understand what something like this represents (if anything), or if  this is just a theoretical construct to prove some larger point.  What does this mean?
Well, let's say my salary is modeled by $f(t)=t^2$
That means the instantaneous rate of change (derivative) of my salary is modeled by $g(t)=2t$
So, what exactly does the below expression represent?   Definite integral calculates the area under the rate curve on [2,3] = the total salary change over [2,3]?  So, the salary grew by 5 on [2,3]
$$\int_2^32tdt=[3^2-2^2]=5$$
This means area under the rate curve on [2,x] = the total salary change over [2,x]?
$$\int_2^x2tdt=??$$
But now, what does this expression below represent?   Why would you take the derivative of the above?  Also, should it be $\frac{d}{dt}$ or $\frac{d}{dx}$ ?  Is the below the rate of change of the total salary change?  
$$\frac{d}{dt}\int_2^x2tdt=??$$
 A: $${d\over dt}\int_1^2 t^2\,dt={d\over dt}\left[{t^3\over 3}\right]\Bigg|_{t=1}^{t=2}={d\over dt}\left[{2^3\over 3}-{1^3\over 3}\right]={d\over dt}\left[{7\over 3}\right]=0.$$
This should not be surprising since $\int_1^2 t^2\,dt$ is a constant, so ${d\over dt}\int_1^2 t^2\,dt={d\over dt}(\text{constant})=0$.
On the other hand, ${d\over dt}\int_1^t x^2\,dx$ is a different animal, but one that makes perfect sense calculationally since
$${d\over dt}\int_1^t x^2\,dx={d\over dt}\left[{x^3\over 3}\right]\Bigg|_{x=1}^{x=t}={d\over dt}\left[{t^3\over 3}-{1^3\over 3}\right]=t^2,$$
(of course, we could have appealed to the other part of the Fundamental Theorem of Calculus to get this immediately, but that is a different point) and this says that the (instantaneous) rate of change of the function $A(t):=\int_1^t x^2\,dx$ is $A'(t)=t^2$. Functions of this form $A(t)$ are sometimes called accumulator functions since they are tracking, in this case, the accumulated (signed) area under $y=x^2$ from $x=1$ to $x=t$ (which is varying).
More generally, for
$$
A(t):=\int_a^t f(x)\,dx,
$$
then 
$$
A'(t)={d\over dt}\int_a^t f(x)\,dx,
$$
represents the rate of change of the accumulated area under $f(x)$ from $a\le x\le t$.
We can ask all the usual calculus question about this "new" class of functions: when is the accumulated area increasing? decreasing? when it is maximum/minimum?
