Fundamental theorem of calculus/Riemann/Section

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Let denote a real interval, let

denote a Riemann-integrable function, and let . Then the function

is called the integral function for for the starting point .

This function is also called the indefinite integral.

The from the theorem is in the animation, and in the theorem is the moving in the animation. The moving point in the animation is a point which exists by the mean value theorem of definite integrals, applied to and .

The following statement is called Fundamental theorem of calculus.

Let denote a real interval, and let

denote a continuous function. Let , and let

denote the corresponding integral function. Then is differentiable, and the identity

holds for all .

Let be fixed. The difference quotient is

We have to show that for , the limit exists and equals . Because of the Mean value theorem for definite integrals, for every , there exists a with

and therefore

For , converges to , and because of the continuity of , also converges to .