The fundamental theorem(s) of calculus relate derivatives and integrals with one another. These relationships are both important theoretical achievements and pactical tools for computation. While some authors regard these relationships as a single theorem consisting of two "parts" (e.g., Kaplan 1999, pp. 218-219), each part is more commonly referred to individually.
While terminology differs (and is sometimes even transposed, e.g., Anton 1984), the most common formulation (e.g., Apostol 1967, p. 202) considers the first
fundamental theorem of calculus, also termed "the fundamental theorem, part
I" (e.g., Sisson and Szarvas 2016, p. 452), to state that, for 
a real-valued continuous
function on an open interval 
and 
any number in 
, if 
is defined by
 |
(1)
|
then
 |
(2)
|
at each number in 
.
Similarly, the most common formulation (e.g., Apostol 1967, p. 205) of the second fundamental theorem of calculus,
also termed "the fundamental theorem, part II" (e.g., Sisson and Szarvas
2016, p. 456), states that if 
is a real-valued continuous
function on the closed interval ![[a,b]](/images/equations/FundamentalTheoremsofCalculus/Inline8.svg)
and 
is the indefinite integral
of 
on ![[a,b]](/images/equations/FundamentalTheoremsofCalculus/Inline11.svg)
,
then
 |
(3)
|
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.
A third fundamental theorem of calculus applies to integrals along curves (i.e., path integrals) and states that if 
has a continuous indefinite integral 
in a region 
containing a parameterized curve 
for 
, then
 |
(4)
|
(Krantz 1999, p. 22).
