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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 is nowadays called 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 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 $x_{1}$ and $x_{1}+\Delta x$ in $[a,b]$, one has $${\begin{aligned}F(x_{1}+\Delta x)-F(x_{1})&=\int _{a}^{x_{1}+\Delta x}f(t)\,dt-\int _{a}^{x_{1}}f(t)\,dt\\&=\int _{x_{1}}^{x_{1}+\Delta x}f(t)\,dt,\end{aligned}}$$ 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

Statement

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 $$\int _{a}^{b}f(x)\,dx=F(b)-F(a).$$

Proof

This is a limit proof by Riemann sums.

To begin, we recall the mean value theorem. Stated briefly, if F is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists some c in (a, b) such that $$F'(c)(b-a)=F(b)-F(a).$$

Let f be (Riemann) integrable on the interval [a, b], and let f admit an antiderivative F on (a, b) such that F is continuous on [a, b]. Begin with the quantity F(b) − F(a). Let there be numbers x₀, ..., x_n such that $$a=x_{0}<x_{1}<x_{2}<\cdots <x_{n-1}<x_{n}=b.$$

It follows that $$F(b)-F(a)=F(x_{n})-F(x_{0}).$$

Now, we add each F(x_i) along with its additive inverse, so that the resulting quantity is equal: $${\begin{aligned}F(b)-F(a)&=F(x_{n})+[-F(x_{n-1})+F(x_{n-1})]+\cdots +[-F(x_{1})+F(x_{1})]-F(x_{0})\\&=[F(x_{n})-F(x_{n-1})]+[F(x_{n-1})-F(x_{n-2})]+\cdots +[F(x_{2})-F(x_{1})]+[F(x_{1})-F(x_{0})].\end{aligned}}$$

The above quantity can be written as the following sum:

$$F(b)-F(a)=\sum _{i=1}^{n}[F(x_{i})-F(x_{i-1})].$$ 1'

The function F is differentiable on the interval (a, b) and continuous on the closed interval [a, b]; therefore, it is also differentiable on each interval (x_i−1, x_i) and continuous on each interval [x_i−1, x_i]. According to the mean value theorem (above), for each i there exists a $c_{i}$ in (x_i−1, x_i) such that $$F(x_{i})-F(x_{i-1})=F'(c_{i})(x_{i}-x_{i-1}).$$

Substituting the above into (1'), we get $$F(b)-F(a)=\sum _{i=1}^{n}[F'(c_{i})(x_{i}-x_{i-1})].$$

The assumption implies $F'(c_{i})=f(c_{i}).$ Also, $x_{i}-x_{i-1}$ can be expressed as $\Delta x$ of partition $i$.

$$F(b)-F(a)=\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].$$ 2'

A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function.

We are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the mean value theorem, describes an approximation of the curve section it is drawn over. Also $\Delta x_{i}$ need not be the same for all values of i, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with n rectangles. Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral. We know that this limit exists because f was assumed to be integrable. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

So, we take the limit on both sides of (2'). This gives us $$\lim _{\|\Delta x_{i}\|\to 0}F(b)-F(a)=\lim _{\|\Delta x_{i}\|\to 0}\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].$$

Neither F(b) nor F(a) is dependent on $\|\Delta x_{i}\|$, so the limit on the left side remains F(b) − F(a). $$F(b)-F(a)=\lim _{\|\Delta x_{i}\|\to 0}\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].$$

The expression on the right side of the equation defines the integral over f from a to b. Therefore, we obtain $$F(b)-F(a)=\int _{a}^{b}f(x)\,dx,$$ which completes the proof.

Generalizations

The function $f$ does not have to be continuous over the whole interval. The first theorem may be applied in the case of Lebesgue integrable function. This concludes 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. The Lebesgue integrable function $f$ may also be applied in the second theorem, which has an antiderivative $F$ but not all integrable functions do. 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. 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.

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.

There is a version of the theorem for complex functions: suppose $U$ is an open set in $\mathbb {C}$ and $f\colon U\to \mathbb {C}$ is a function that has a holomorphic antiderivative $F$ on $U$. Then for every curve $\gamma \colon [a,b]\to U$, 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 $\partial M$ 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. $\mathbb {R} ^{k}$ 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.