Visual Calculus - Fundamental Theorem of Calculus

Theorem. Let f be a function which is continuous on the interval [a, b].

Let F be an indefinite integral or antiderivative of f. Then
The function
is an indefinite integral or antiderivative of f. That is,
A'(x) = f(x)

The first part of this theorem tells us how to evaluate a definite integral provided that f has an indefinite integral.

The second part of the theorem tells us that f has an indefinite integral. The only remaining problem is actually finding a formula for the indefinite integral which we can easily evaluate.

Look at some examples applying the first part of the theorem.
Some drill problems applying the first part of the theorem.
[Using Java] [Using IBM TechExplorer] [Using IBM Pro. TechExplorer]
A simple example
[Using Flash] [Using Java]
An interactive Java applet illustrating the function A in the second part of the theorem.
An animation similar to the preceding.
[Using animated gifs] [Using LiveMath]
Illustration of the Fundamental Theorem of Calculus using Maple and a LiveMath Notebook.

In general, we will not be able to find a "formula" for the indefinite integral of a function. However, using the second part of the Fundamental Theorem, we are still able to draw the graph of the indefinite integral:

Graphing the function A from the second part of the theorem using a graphing calculator or a LiveMath Notebook.

Check out the proof of the Fundamental Theorem:

proof of part two
[Using Flash] [Using Java]
proof of part one
[Using Flash] [Using Java]

The fundamental theorem of calculus and the chain rule:

Example 1.
[Using Flash] [Using Java]
Example 2.
[Using Flash] [Using Java]
LiveMath Notebook which evaluates the derivative of a function which is an integral with variable limits.