Fundamental Theorems of Calculus

The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , then

(1)

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.

The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by

(2)

then

(3)

at each point in .

The fundamental theorem of calculus along curves states that if has a continuous indefinite integral in a region containing a parameterized curve for , then

(4)

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