Fundamental theorem of calculus

A pair of results showing that differentiation and integration are inverse operations: definite integrals measure accumulated change, and derivatives recover the rate of accumulation.

First fundamental theorem of calculus: A theorem stating that the derivative of an accumulation function equals its continuous integrand. It formalizes how differentiating accumulated change recovers the original rate.
Derivative: The instantaneous rate of change of a function, defined through a limit of difference quotients. The theorem connects this local rate to integral accumulation.
Isaac Newton: An English mathematician and natural philosopher who developed calculus in the seventeenth century. His method of fluxions used the link between rates of change and accumulated quantities.
Displacement (vector): The change in position of an object, represented by a vector from initial to final position. Integrating velocity gives displacement over a time interval.
Mean value theorem for integrals: A theorem stating that a continuous function's average value on an interval equals one of its values there. It links integrals to pointwise values but does not assert that integration and differentiation are inverses.
Second fundamental theorem of calculus: A theorem that evaluates a definite integral using an antiderivative of its integrand. It turns accumulated change over an interval into an endpoint difference.
Definite integral: A signed accumulation of a function over an interval, defined through limiting sums. This is the accumulated quantity that the theorem links to derivatives.
Gottfried Wilhelm Leibniz: A German philosopher and mathematician who independently developed calculus and its notation. His integral notation helped express the accumulation side of the theorem.
Work (physics): Energy transferred when a force causes displacement, measured by integrating force along a path. The theorem relates a variable force's accumulated work to an antiderivative's endpoint values.
Improper integral: An integral defined by a limit when an interval is unbounded or the integrand is unbounded. The standard theorem's hypotheses do not directly cover these limiting cases.

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