There is one small thing that we have not dealt with yet with respect to getting the exact value of the net change of a function (i.e., $\int _a^b f'(t)\, dt = f(b) - f(a)$). Although it seems obvious from the definition of “net change” (from Integrated Language), as well as our estimate calculations with rectangles (rate times change in time- from Winter Storm Warning), we haven’t officially shown that the value of the integral (and finding the area between $f'(t)$ and the $t$-axis) is indeed exactly the same as the change of the anti-derivative of $f'(t)$ (rather than just stating that it is like we did in Integrated Language) and the area under the graph of $f'(t)$. That is, if we can find the function whose derivative is the rate function, finding the exact net change (and the exact area) is just two “plug-n-chugs” away!

We will show this here with a geometrical argument:

We have a graph of $f'(t)$ here. We want to find the area between the graph and the $t$-axis on the interval $(a, b)$ (Figure 1). To do so, we define the “window-wiper” area function $A(t)$. $A(t)$ is the area under the curve from $0$ to $t$. Thus, what does $A(a)$ represent? What does $A(b)$ represent? Write the area that we want in terms of $A(t)$ (Hint: A difference of values of $A(t)$ at two times). Thus, our problem comes down to showing that $A(t)$ is the same as $f(t)$ (or, equivalently, that $A'(t)$ is the same as $f'(t)$).

Consider two times within the interval $(a, b)$ with the times being very close to each other. Call them $t$ and $t+h$, where $h$ is very small. We’d like the area underneath the curve of $f'(t)$ for this interval of times only (Figure 2). What would this area be in terms of $A(t)$? (Hint: Another difference). On the other hand, since $h$ is very small, this same small, skinny region looks very much like what shape? What is the area of that shape (in terms of $h$ and $f'(t)$)?

Thus, we have two expressions representing the very same area of the very same (skinny) region. Thus, set them equal to each other. Now divide both sides by $h$. For very small $h$, what does the side involving $A(t)$ represent? What must $A'(t)$ indeed be the same as? Thus, what must $A(t)$ be the same as? Thus, what is $A(b) - A(a)$ the same as (in terms of $f$)?

Thus, we now have hope of finding the exact net change in an amount function- if only we can find the antiderivative of the rate function!