Calculus in the ancient times: Method of exhaustion (see wikipedia)

User-friendly definition of the integral:

It would be better if the integral would not depend on the definition of the bins. In the limit it doesn’t matter if they are evenly spaced or not:

Fundamental theorem of calculus:

If is a continuous real-valued function on a closed interval

then is continuous and differentiable in and:

as a corollary:

### Exercises

(tip: by parts)

Try to solve a more general formula:

(get the solution using Wolfram:alpha )

For the die-hard:

( solution )

tip: make a substitution deplyoing this relationship:

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