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: and . Let’s see if we can explain this. Consider the following graph:An accumulation function measures the signed area in the region between and the
-axis. Hence is playing the role of a “place-holder” that allows us to evaluate . On
the other hand, is the **specific number** that we are using to bound the region that
will determine the area between and the -axis, and hence the value of .

However when , starts to accumulate positively signed area, and hence is increasingdecreasing . Thus is increasing on , decreasing on and hence has a local minimum at .

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

### The First Fundamental Theorem of Calculus

Let be a continuous function on the real numbers and consider From our previous work we know that is increasing when is positive and is decreasing when is negative. Moreover, with careful observation, we can even see that is concave up when is positive and that is concave down when 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 and the function . This is a good guess, check out our next theorem:

The First Fundamental Theorem of Calculus says that an accumulation function of is an antiderivative of . Another way of saying this is:**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.

Since , it follows that the function is increasing on the interval and decreasing on the interval Since the function , the derivative of , is increasing on , is concave up on the interval Since is constant on the interval , is linear on .

The function has a local and global minimum at and the global maximum at Now we are ready to sketch the graph of , on the same set of axes as the graph of .

What if the function is the velocity function for an object moving along a straight line, i.e. , ?

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

We use different variables in that case, since we want to be a function of time, . With this adjustment, we define the accumulation function as follows. Since the integral gives the displacement of the object on the time interval , it follows that where gives the position of the object at the time . If we differentiate this equation with respect to , we get that Since , we have that This is the the First Fundamental Theorem of Calculus!