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The rate that accumulated area under a curve grows is described identically by that curve.

### Accumulation functions

While the definite integral computes a signed area, which is a fixed number, there is a way to turn it into a function.

One thing that you might notice is that an accumulation function seems to have two variables: $x$ and $t$. Let’s see if we can explain this. Consider the following graph:

An accumulation function $F$ measures the signed area in the region $[a,x]$ between $f$ and the $t$-axis. Hence $t$ is playing the role of a “place-holder” that allows us to evaluate $f$. On the other hand, $x$ is the specific number that we are using to bound the region that will determine the area between $f$ and the $t$-axis, and hence the value of $F$.

Given what is $F(5)$?
What is $F(-5)$?
What is $F(-3)$?

Working with the accumulation function leads us to a question, what is when $x< a$? The general convention is that With this in mind, let’s consider one more example.

The key point to take from these examples is that an accumulation function is increasing precisely when $f$ is positive and is decreasing precisely when $f$ is negative. In short, it seems that $f$ is behaving in a similar fashion to $F'$.

### The First Fundamental Theorem of Calculus

Let $f$ be a continuous function on the real numbers and consider From our previous work we know that $F$ is increasing when $f$ is positive and $F$ is decreasing when $f$ is negative. Moreover, with careful observation, we can even see that $F$ is concave up when $f'$ is positive and that $F$ is concave down when $f'$ is negative. Thinking about what we have learned about the relationship of a function to its first and second derivatives, it is not too hard to guess that there must be a connection between $F'$ and the function $f$. This is a good guess, check out our next theorem:

The First Fundamental Theorem of Calculus says that an accumulation function of $f$ is an antiderivative of $f$. Another way of saying this is: This could be read as:

The rate that accumulated area under a curve grows is described identically by that curve.

Now that we are working with accumulation functions, let’s see what happens when we compose them with other functions.

Let’s practice this once more.

What if the function $f$ is the velocity function for an object moving along a straight line, i.e. $f(t)=v(t)$, $a\le t\le b$?

What is the meaning of an accumulation function in that case?

We use different variables in that case, since we want $F$ to be a function of time, $t$. With this adjustment, we define the accumulation function $F$ as follows. Since the integral gives the displacement of the object on the time interval $[a,t]$, it follows that where $s(t)$ gives the position of the object at the time $t$. If we differentiate this equation with respect to $t$, we get that Since $s'(t)=v(t)$, we have that This is the the First Fundamental Theorem of Calculus!