The Accumulation Function and the FTC

What is the connection between the area under a function curve and the antiderivative of that function?
  1. When the app is first started (or reset), the function is shown (in red). is defined to be . This means that the value of is the same as the area under between a fixed point and another point .
  2. By the FTC, . Check the "Show f(t)" box to display , the derivative of (in blue).
  3. You can drag the point on the black cursor on the function (red) to analyze its derivative (blue), such as finding sign changes of at extrema of .
  4. We can also use a function graph to analyze its antiderivative .
  5. Check the "Show FTC" box. Do not move point "a" at this time.
  6. Drag the point x so that it coincides with a. What is the area under between a and x? What is the value of at this point?
  7. Drag x to various points left and right of a. The signed area under between a and x is the -value of at x. This is because the area under the graph of a function's rate of change on an interval (for example, velocity) gives the amount of change of the function (position) on that interval. For our example, we could say that : the area under from a to x is equal to the change in the value of from a to x.
  8. Now leave x fixed, and move the point a along the x-axis. What do you observe? This happens because changing the value of a changes the constant of integration of the function .
is an antiderivative of . 's -value at any value of x is equal to the area under between a and x, which is . So both a and x affect the value of . In this sense, is a function of x, and we write . Notice the use of the variable in the integral. Using the value of x as a limit of integration gives it a different role than the variable of integration, so we have to use a different variable inside the integral. As we know, however, the variable name we choose makes no difference in a definite integral. We just can't use the same variable for two different purposes. If we fix x and change a, we again change the area between them by a certain amount. (The amount is the area between the old and new positions of a). This amount added to shifts the graph up or down. Thus, a determines the value of the "" in the antiderivative. is "an" antiderivative of , whose value depends on a. So no matter what the value of a is, the derivative of is just , since the derivative of any value of is zero. This leads to the following equation, which is the Antiderivative Part of the Fundamental Theorem of Calculus: