# First Fundamental Theorem of Calculus

- Author:
- Dr. Jack L. Jackson II

- Topic:
- Calculus

## Accumulation Function

We suggest that you first explore the Second Fundamental Theorem of Calculus via Dr. Jackson's GeoGebra activity before proceeding. https://www.geogebra.org/m/cn2khzjf

## Differentiating and then Integrating

In the App
Start by typing in any formula for a function ?

*f*(*x*) in the input box. If you check the f(x) checkbox in the right window the graph of*f*(*x*) will appear in the right window in blue. The formula for f '(x) is displayed, along with the graph of*f*'(*x*) in red in the left window. For now, toggle off the graph of*f*(*x*) to clear out most of the right window. Choose a value for*a*via the slider or input box in the left window. Similarly pick a value for*x*. Start the value of*x*the same as the value for a and slowly slide the slider for*x*to the right. You will see area accumulating between the graph of*f*'(*x*) and the*x*-axis. Green areas accumulate positively and red areas accumulate negatively. The green area minus the red area is the value of the accumulation function for that value of*x*. Click the checkbox for (*x*,*A*(*x*)) in the right window to see this value graphed there. Move*x*around via the slider to see it change. Now click on the checkbox for*A*(*x*) to see the graph. Again move*x*around to investigate. Now deselect (*x*,*A*(*x*)) to hide that portion of the illustration. Does the graph of*A*(*x*) look familiar? Select the checkbox for*f*(*x*). How does the graph of*A*(*x*) compare to the graph of*f*(*x*)? Does they look like vertical shifts of each other? Select the checkbox for Shift in the right window. This will show how far apart the two graphs are vertically for a particular*x*-value. Move this point on the graph of*f*(*x*) around. Does the vertical distance stay constant? How does this distance compare to*f*(*x*)? You should see that, yes, the graphs of*A*(*x*) and*f*(*x*) are vertical shifts of each other and that the amount of the vertical shift is*f*(*a*). What does this tell use about an alternate way to express## First Fundamental Theroem of Calculus

(Fundamental Theorem of Calculus Part 1)
If f is any function differentiable on the interval including a and b and any points between them, then
.
First differentiating and then integrating produces the original function, possibly with a vertical shift.
One of the consequences of this is that if we are integrating a function .

*g*(*x*) and we can find a function*f*(*x*) so that*g*(*x*) =*f*'(*x*) (i.e. we find any antiderivative*f*for the original function*g*), then we can find an exact value for the definite integral by finding the total change in the antiderivative over the interval: