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Background

Discovery

The fundamental theorem of calculus relates to differentiation and integration, showing that these 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 knew how to compute area via infinitesimals, 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 closely related.

From the conjecture and the proof of the fundamental theorem of calculus, calculus as a 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. Isaac Barrow proved a more generalized version of the theorem. His student Isaac Newton completed the development of the surrounding mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.

Overview

Main articles: Derivative, Riemann sum, and Definite integral

The derivative of a function with a single variable is a tool quantifying the sensitivity of change of a function's output concerning its input. When it exists, it can be considered as the tangent line's slope to the graph of the function at that point. Given that a function of a real variable $f(x)$ is differentiable in real domain, the derivative of a function $f$ with respect to $x$, denoted as $f'(x)$, can be defined in terms of limit: $$f'(x)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.$$

The integral is a continuous summation, used in calculating the area under a graph and the volume of a graph revolving around an axis. Such calculations are implemented whenever there are two points bounded in the real line called the interval exists, and the integral is called the definite integral. This integral is defined by using Riemann sum: given that $f$ differentiable at $[a,b]$, and partition of such interval $P$ that can be expressed as $a=x_{0}<x_{1}<\dots <x_{n}=b$, then the definite integral can be defined as $$\int _{a}^{b}f(x)\,dx=\lim _{\|P\|\to 0}\sum _{i=1}^{n}f({\bar {x}}_{i})\Delta x_{i},$$ where $\Delta x_{i+1}$ represents the difference between two each $x_{i+1}$ and $x_{i}$ in the interval.

First theorem

Statement

The first fundamental theorem of calculus describes the value of any function as the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point. It can be interpreted as an example that velocity is the function, and integrating it from the starting time up to any given time to obtain a distance function whose derivative is that velocity. The first fundamental theorem of calculus is stated formally as follows: let $f$ be a continuous real-valued function defined on a closed interval $[a,b]$. For all $x$ in that same closed interval, let $F$ be the function defined as: $$F(x)=\int _{a}^{x}f(t)\,dt.$$ Then $F$ is continuous on $[a,b]$ and differentiable on the open interval $(a,b)$, and $$F'(x)=f(x)$$ for all $x$ in $(a,b)$, such that $F$ is an antiderivative of $f$.

Proof

For a given function f, define the function F(x) as $$F(x)=\int _{a}^{x}f(t)\,dt.$$ For any two numbers x₁ and x₁ + Δx in [a, b], we have $${\begin{aligned}F(x_{1}+\Delta x)-F(x_{1})&=\int _{a}^{x_{1}+\Delta x}f(t)\,dt-\int _{a}^{x_{1}}f(t)\,dt\\&=\int _{x_{1}}^{x_{1}+\Delta x}f(t)\,dt,\end{aligned}}$$ The latter equality results from the basic properties of integrals and the additivity of areas.

According to the mean value theorem for integration, there exists a real number $c\in [x_{1},x_{1}+\Delta x]$ such that $$\int _{x_{1}}^{x_{1}+\Delta x}f(t)\,dt=f(c)\cdot \Delta x.$$ It follows that $$F(x_{1}+\Delta x)-F(x_{1})=f(c)\cdot \Delta x,$$ and thus that $${\frac {F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}}=f(c).$$ Taking the limit as $\Delta x\to 0,$ and keeping in mind that $c\in [x_{1},x_{1}+\Delta x],$ one gets $$\lim _{\Delta x\to 0}{\frac {F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}}=\lim _{\Delta x\to 0}f(c),$$ that is, $$F'(x_{1})=f(x_{1}),$$ according to the definition of the derivative, the continuity of f, and the squeeze theorem.

Second theorem

The second fundamental theorem says that the sum of infinitesimal changes in a quantity (the integral of the derivative of the quantity) adds up to the net change in the quantity. Formally, let $f$ be a real-valued function on a closed interval $[a,b]$ and $F$ a continuous function on $[a,b]$ which is an antiderivative of $f$ in $(a,b)$: $$F'(x)=f(x).$$ If $f$ is Riemann integrable on $[a,b]$, then $$\int _{a}^{b}f(x)\,dx=F(b)-F(a).$$

Generalizations

The function f does not have to be continuous over the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable function on [a, b] and x₀ is a number in [a, b] such that f is continuous at x₀, then $$F(x)=\int _{a}^{x}f(t)\,dt$$

is differentiable for x = x₀ with F′(x₀) = f(x₀). We can relax the conditions on f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function F is differentiable almost everywhere and F′(x) = f(x) almost everywhere. On the real line this statement is equivalent to Lebesgue's differentiation theorem. These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions.

In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every x, the average value of a function f over a ball of radius r centered at x tends to f(x) as r tends to 0.

Part II of the theorem is true for any Lebesgue integrable function f, which has an antiderivative F (not all integrable functions do, though). In other words, if a real function F on [a, b] admits a derivative f(x) at every point x of [a, b] and if this derivative f is Lebesgue integrable on [a, b], then $$F(b)-F(a)=\int _{a}^{b}f(t)\,dt.$$

This result may fail for continuous functions F that admit a derivative f(x) at almost every point x, as the example of the Cantor function shows. However, if F is absolutely continuous, it admits a derivative F′(x) at almost every point x, and moreover F′ is integrable, with F(b) − F(a) equal to the integral of F′ on [a, b]. Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F′ = f almost everywhere.

The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F(x) admits a derivative f(x) at all but countably many points, then f(x) is Henstock–Kurzweil integrable and F(b) − F(a) is equal to the integral of f on [a, b]. The difference here is that the integrability of f does not need to be assumed.

The version of Taylor's theorem that 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 U is an open set in C and f : U → C is a function that has a holomorphic antiderivative F on U. Then for every curve γ : [a, b] → U, the curve integral can be computed as $$\int _{\gamma }f(z)\,dz=F(\gamma (b))-F(\gamma (a)).$$

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifolds. One such generalization offered by the calculus of moving surfaces is the time evolution of integrals. The most familiar extensions of the fundamental theorem of calculus in higher dimensions are the divergence theorem and the gradient theorem.

One of the most powerful generalizations in this direction is the generalized Stokes theorem (sometimes known as the fundamental theorem of multivariable calculus): Let M be an oriented piecewise smooth manifold of dimension n and let $\omega$ be a smooth compactly supported (n − 1)-form on M. If ∂M denotes the boundary of M given its induced orientation, then $$\int _{M}d\omega =\int _{\partial M}\omega .$$

Here d is the exterior derivative, which is defined using the manifold structure only.

The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold (e.g. R^k) on which the form $\omega$ is defined.

The fundamental theorem of calculus allows us to pose a definite integral as a first-order ordinary differential equation. $$\int _{a}^{b}f(x)\,dx$$ can be posed as $${\frac {dy}{dx}}=f(x),\;\;y(a)=0$$ with $y(b)$ as the value of the integral.