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## Fundamental theorem of calculusThefundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. This means that if a function is first integrated and then differentiated, the original function is retrieved. An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
## IntuitionIntuitively, the theorem simply says that if you know all the little instantaneous changes in some quantity, then you may compute the overall change in the quantity by adding up all the little changes.
To get a feeling for the statement, we will start with an example. Suppose you travel in a straight line, starting at time ## Formal statementsStated formally, the theorem says:
If the function
Here,
Part II of the theorem gives an important method for computing the integral of the continuous function
If is As an example, suppose you need to calculate
## GeneralizationsWe don't need to assume continuity off on the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable function on and is a number in such that is continuous at , then
Part II of the theorem is true for any Lebesgue integrable function The version of Taylor's theorem which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem.
There is a version of the theorem for complex functions: suppose The most powerful statement in this direction is Stokes' theorem. | |||||||

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