The fundamental theorem of calculus is a
theorem that links the concept of
differentiating a
function (calculating its
slopes, or rate of change at each time) with the concept of
integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.
The first part of the theorem, the first fundamental theorem of calculus, states that for a function , an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
or indefinite integral may be obtained as the integral of over an interval with a variable upper bound. This implies the existence of antiderivatives for
continuous functions.
Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed
interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by
symbolic integration
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that
:\frac = f(x).
This is a ...
, thus avoiding
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equatio ...
.
History
The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially
inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient
Greek mathematicians
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
* Greeks, an ethnic group.
* Greek language, a branch of the Indo-European language family.
** Proto-Greek language, the assumed last common ances ...
knew how to compute area via
infinitesimals
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
, an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of ''
continuity'' of functions and ''
motion'' were studied by the
Oxford Calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.
From the conjecture and the proof of the fundamental theorem of calculus,
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
as an unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character,
was by
James Gregory (1638–1675).
Isaac Barrow (1630–1677) proved a more generalized version of the theorem,
while his student
Isaac Newton (1642–1727) completed the development of the surrounding mathematical theory.
Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced
the notation used today.
Geometric meaning
The first fundamental theorem may be interpreted as follows. For a
continuous function whose graph is plotted as a curve, each value of has a corresponding area function , representing the area beneath the curve between and . The area may not be easily computable, but it is assumed to be well-defined.
The area under the curve between and could be computed by finding the area between and , then subtracting the area between and . In other words, the area of this "strip" would be .
There is another way to ''estimate'' the area of this same strip. As shown in the accompanying figure, is multiplied by to find the area of a rectangle that is approximately the same size as this strip. So:
In fact, this estimate becomes a perfect equality if we add the red "Excess" area in the diagram. So:
Rearranging terms:
As approaches in the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, the last fraction must go to zero. To see this, note that the excess region is inside the tiny black-bordered rectangle, giving an upper bound for the excess area:
where
and
are points where reaches its maximum and its minimum, respectively, in the interval .
Thus:
By the continuity of , the right-hand expression tends to zero as does. Therefore, the left-hand side also tends to zero, and:
That is, the derivative of the area function exists and is equal to the original function , so the area function is an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of the original function.
Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are
inverse operations which reverse each other. This is the essence of the Fundamental Theorem.
Physical intuition
Intuitively, the fundamental theorem states that ''integration'' and ''differentiation'' are essentially inverse operations which reverse each other.
The second fundamental theorem says that the sum of
infinitesimal changes in a quantity over time (the integral of the derivative of the quantity) adds up to the net change in the quantity. To visualize this, imagine traveling in a car and wanting to know the distance traveled (the net change in position along the highway). You can see the velocity on the speedometer but cannot look out to see your location. Each second, you can find how far the car has traveled using , multiplying the current speed (in kilometers or miles per hour) by the time interval (1 second =
hour). Summing up all these small steps, you can calculate the ''total'' distance traveled, without ever looking outside the car:
As
becomes
infinitesimally small, the summing up corresponds to
integration. Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position).
The first fundamental theorem says that any quantity is the rate of change (the derivative) of the integral of the quantity from a fixed time up to a variable time. Continuing the above example, if you imagine a velocity function, you can integrate it from the starting time up to any given time to obtain a distance function whose derivative is the given velocity. (To obtain the highway-marker position, you need to add your starting position to this integral.)
Formal statements
There are two parts to the theorem. The first part deals with the derivative of an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
, while the second part deals with the relationship between antiderivatives and
definite integrals.
First part
This part is sometimes referred to as the ''first fundamental theorem of calculus''.
Let be a continuous
real-valued function defined on a
closed interval . Let be the function defined, for all in , by
Then is
uniformly continuous on and differentiable on the
open interval , and
for all in so is an antiderivative of .
Corollary
The fundamental theorem is often employed to compute the definite integral of a function
for which an antiderivative
is known. Specifically, if
is a real-valued continuous function on