Fundamental theorem of calculus proof

This section of Calculus math help discusses the fundamental theorem of calculus proof. There are formal proofs of the fundamental theorem of calculus and the informal proof of the fundamental theorem of calculus. This section deals with the informal fundamental theorem of calculus proof so that calculus students may understand how the fundamental theorem of calculus was derived.

Recall the definition of the derivative

By the definition of the derivative:

exists and that this limit equals f(x).

x and x increment illustration

For illustration of the fundamental theorem of calculus proof, lets assume that the function f9x0 has positive values on the interval [a,b] so that an integral can be thought of as representing an area under the graph of f. Think of x as representing some point between a and b. Think of h as representing some small positive number such that x + h is also between a and b and is a little bit to the right of x. The picture of this illustration is shown below.

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