Fundamental Theorem of Calculus

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The fundamental theorem of calculus consists (intuitively) in the statement that the differentiation and integration of a function are inverse operations. This means that any bounded and integrable function (being continuous or discontinuous in a finite number of points) verifies that the derivative of its integral is equal to itself. This theorem is central to the branch of mathematics called mathematical analysis or infinitesimal calculus.

The theorem was fundamental because until then the approximate calculation of areas -integrals- that had been worked on since Archimedes, was a branch of mathematics that was followed separately from the differential calculus that had been developed by Isaac Newton, Isaac Barrow and Gottfried Leibniz in the 18th century, and gave rise to concepts such as derivatives. Integrals were investigated as ways of studying areas and volumes, until at that point in history both branches converged, when it was shown that the study of the "area under a function" was closely linked to differential calculus, resulting in integration being the inverse operation to the shunt.

A direct consequence of this theorem is Barrow's rule, sometimes called the second fundamental theorem of calculus, and which allows us to calculate the integral of a function using the indefinite integral of the function, being integrated.

History

The fundamental theorem of calculus concerns differentiation and integration, showing that these two operations are essentially inverses of each other. Before the discovery of this theorem, it was not recognized that these two operations were related. The ancient Greek mathematicians knew how to calculate area through the infinitesimals, an operation we would now call integration. The origins of differentiation also predate the Fundamental Theorem of Calculus by hundreds of years; for example, in the XIV century the notions of continuity of functions and of motion were studied by Oxford calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the verification that these two apparently different operations (calculation of geometric areas and calculation of speeds) were ultimately closely related.

The first published statement and proof of a restricted version of the fundamental theorem was made by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while Barrow's student, Isaac Newton (1642–1727), completed the development of the mathematical theory concerned. Gottfried Leibniz (1646–1716) systematized knowledge in a calculation of infinitesimal quantities and introduced the notation used today.

Geometric intuition

The area in red can be calculated as

h

Sometimes

f(x)

or, if the function was known

{displaystyle A(x)}

Like,

{displaystyle A(x+h)-A(x)}

. These values are approximately equal, especially for small values

h

Suppose you have a continuous function ${displaystyle y=f(x)}$ whose graphic representation is a curve. So, for every value of $x$ makes sense intuitively to think that there is a function ${displaystyle A(x)}$ which represents the area under the curve ${displaystyle 0}$ and $x$ even without knowing his expression.

Suppose now you want to calculate the area under the curve $x$ and ${displaystyle x+h}$ . It could be done by finding the area between ${displaystyle 0}$ and ${displaystyle x+h}$ and then subtract the area between ${displaystyle 0}$ and $x$ . In short, the area would be ${displaystyle A(x+h)-A(x)}$ .

Another way to estimate this same area is to multiply $h$ for $f(x)$ to find the area of a rectangle that matches roughly the "loach". Note that the approximation to the sought area is more precise the smaller the value of $h$ .

Therefore, you can say that ${displaystyle A(x+h)-A(x)}$ is approximately equal to ${displaystyle f(x)cdot h}$ and that the accuracy of this approach improves by diminishing the value of $h$ . In other words, ${displaystyle f(x)cdot happrox A(x+h)-A(x)}$ becoming this approach to equality when $h$ tends to 0 as a limit.

Splitting the two sides of the equation by $h$ obtained

${displaystyle f(x)approx {frac {A(x+h)-A(x)}{h}}}$

When $h$ tends to ${displaystyle 0}$ , it is observed that the right member of the equation is simply the derivative ${displaystyle A'(x)}$ of the function ${displaystyle A(x)}$ and that the left member stays in $f(x)$ To no longer be $h$ present.

It is then shown informally that ${displaystyle f(x)=A'(x)}$ , that is, the derivative of the area function ${displaystyle A(x)}$ is actually the function $f(x)$ . In other words, the area function ${displaystyle A(x)}$ is the anti-default of the original function.

What has been shown is that, intuitively, calculating the derivative of a function and "finding the area" under its curve are "inverse" operations, that is, the objective of the Fundamental Theorem of Integral Calculus.

First Fundamental Theorem of Calculus

Theorem

Sea $f$ an integrated function in the interval ${displaystyle [a,b],}$ defined $F$ in $[a,b]$ Like

{displaystyle F(x)=int _{a}^{x}f(t)dt}

Yeah. $f$ is continuous $c in (a,b)$ , then $F$ is differential in $c$ and

{displaystyle F'(c)=f(c)}

Motto

Sea $f$ integrable over $[a,b]$ and ${displaystyle mleq f(x)leq Mquad forall ;xin [a,b]}$ then.

m(b-a) leq {int_a^b f(t)dt} leq M(b-a)

Demo

It is clear that

m(b-a)leq L(f,P) hbox{y} U(f,P) leq M(b-a)

for all partitions $P$ . Post

int_{a}^{b}f = sup {L(f,P)}=inf{U(f,P)}

The inequality follows immediately.

Demo 1

By definition you have to

F'(c)={ lim_{h rightarrow 0} {frac{F(c+h)-F(c)}{h}} }

Sea $0}" xmlns="http://www.w3.org/1998/Math/MathML">h▪0{displaystyle h 2005}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbddb7a5cca6170575e4e73e769fbb434c2a3d71" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;"/>$ then.

F(c+h)-F(c)={int_c^{c+h} f(t)dt}

They are defined $m_h$ and $M_h$ like:

{displaystyle {begin{aligned}m_{h}&=inf{f(x)|cleq xleq c+h}\M_{h}&=sup{f(x)|cleq xleq c+h}end{aligned}}}

Applying the lemma it is observed that:

m_h cdot h leq {int_c^{c+h} f(t)dt} leq M_h cdot h

Therefore,

m_h leq frac{F(c+h)-F(c)}{h} leq M_h

Sean. $<math alttext="{displaystyle hh.0{displaystyle h vis0} <img alt="h$ and

{displaystyle {begin{aligned}{m}_{h}^{*}&=inf{f(x)|c+hleq xleq c}\{M}_{h}^{*}&=sup{f(x)|c+hleq xleq c}end{aligned}}}

Applying the lemma it is observed that

{displaystyle {m}_{h}^{*}cdot (-h)leq {int _{c+h}^{c}f(t)dt}leq {M}_{h}^{*}cdot (-h)}

How

F(c+h)-F(c)={int_c^{c+h} f(t)dt} = -{int_{c+h}^{c} f(t)dt}

then

{displaystyle {M}_{h}^{*}cdot hleq F(c+h)-F(c)leq {m}_{h}^{*}cdot h}

Post $<math alttext="{displaystyle hh.0{displaystyle h vis0} <img alt="h$ You have to.

{displaystyle {m}_{h}^{*}leq {frac {F(c+h)-F(c)}{h}}leq {M}_{h}^{*}}

And like $f$ is continuous $c$ You have to

{displaystyle lim _{hrightarrow 0}m_{h}=lim _{hrightarrow 0}M_{h}=lim _{hrightarrow 0}{m}_{h}^{*}=lim _{hrightarrow 0}{M}_{h}^{*}=f(c)}

and this leads to

F'(c)={ lim_{h rightarrow 0} {frac{F(c+h)-F(c)}{h}} } = f(c)

Demo 2

Another demonstration of the fundamental theorem of the calculation
Taking a closed interval $[a,x]$ on $[a,b]$ Since $f(t)$ is continuous $[a,b]$ , it will also be in $[a,x]$ . According to the average value theorem for integrals it is fulfilled that: $exists xiin[a,x] : quad f(xi)= frac{1}{x-a}int_{a}^{x}f(t)dt$ Making the interval very small so that $x longrightarrow a$ and because of that trend, it also has to $xi longrightarrow a$ So in the limits you reach: $lim_{xi to a} f(xi) = lim_{x to a} frac{int_{a}^{x}f(t)dt}{x-a}$ We know that: $int_{a}^{a}f(t)dt = 0$ Then the equation can be written as: $lim_{xi to a} f(xi) = lim_{x to a} frac{int_{a}^{x}f(t)dt - int_{a}^{a}f(t)dt}{x-a}$ Given that $F(x)=int_{a}^{x}f(t)dt$ then. $F(a)=int_{a}^{a}f(t)dt$ $lim_{xi to a} f(xi) = lim_{x to a} frac{F(x) - F(a)}{x-a}$ And because $f(t)$ is continuous in a, then $lim_{xi to a} f(xi) = f(a)$ $f(a) = lim_{x to a} frac{F(x) - F(a)}{x-a}$ View the equation otherwise: $f(x)\|_{x=a} = frac{dF(x)}{dx}\|_{x=a}$ So. $frac{dF(x)}{dx} = f(x)$ or also $frac{d}{dx}int_{a}^{x}f(t)dt = f(x)$ And so on $forall cin(a,b) : quad frac{d}{dx}int_{a}^{x}f(t)dt \|_{x=c} = f(c)$ This shows the first fundamental theorem of the calculation.

Consequences

Corollary

Yeah. $f$ is continuous $[a,b]$ and ${displaystyle f=g'}$ for some function $g$ then.

{displaystyle int _{a}^{b}f(t)dt=g(b)-g(a)}

Demo

Let

{displaystyle F(x)=int _{a}^{x}f(t)dt}

then. ${displaystyle F'=f=g'}$ in $[a,b]$ , so ${displaystyle exists ;cin mathbb {R} }$ such as

{displaystyle F=g+c}

Note that

{displaystyle 0=F(a)=g(a)+c}

from where it follows that ${displaystyle c=-g(a)}$ so;

{displaystyle F(x)=g(x)-g(a)}

In particular when $x=b$ then.

{displaystyle int _{a}^{b}f(t)dt=F(b)=g(b)-g(a)}

Sometimes this corollary is called the «second fundamental theorem of calculus».

If we use the chain rule we obtain as a direct consequence of the first fundamental theorem of calculus

{displaystyle {dfrac {mathop {} !mathrm {d} }{mathop {} !mathrm {d} x}}{int _{a(x)}^{b(x)}f(t)mathop {} !mathrm {d} t}=f(b(x))cdot b^{prime }(x)-f(a(x))cdot a^{prime }(x)}

being ${textstyle f(t)}$ a continuous function on the interval ${displaystyle left[a(x),b(x)right]}$ where ${displaystyle a(x)}$ and ${displaystyle b(x)}$ are different functions.

Examples

Example 1

Yes

{displaystyle F(x)=int _{0}^{x}t^{2}dt}

then

${displaystyle F'(x)=x^{2}}$

Example 2

Yes

{displaystyle H(x)=int _{0}^{e^{3x}}operatorname {sen}(t)dt}

then

{displaystyle {begin{aligned}H'(x)&=operatorname {sen} left(e^{3x}right)e^{3x}cdot 3\&=3e^{3x}operatorname {sen} left(e^{3x}right)end{aligned}}}

Example 3

Yes

{displaystyle G(x)=int _{0}^{x^{2}}{text{arcsen}}(t)dt}

then

{displaystyle {begin{aligned}G'(x)&={text{arcsen}}(x^{2})cdot 2x\&=2x;{text{arcsen}}(x^{2})end{aligned}}}

Example 4

Yes

{displaystyle J(x)=int _{0}^{int _{a}^{x}{frac {1}{(1+operatorname {sen} ^{2}t)}}dt}{frac {1}{(1+operatorname {sen} ^{2}t)}}dt}

then

{displaystyle J'(x)={frac {1}{left(1+operatorname {sen} ^{2}(int _{a}^{x}{frac {1}{(1+operatorname {sen} ^{2}t)}}dt)right)}},cdot ,{frac {1}{(1+operatorname {sen} ^{2}x)}}}

Second Fundamental Theorem of Calculus

The second fundamental theorem of integral calculus (or Newton-Leibniz rule, or also Barrow's rule, after the English mathematician Isaac Barrow, Isaac Newton's teacher) is a property of continuous functions that allows you to easily calculate the value of the definite integral from any of the function's primitives.

Theorem

Sea $f$ an integrated function in the interval $[a,b]$ and ${displaystyle f=g'}$ for some function $g$ then.

{displaystyle int _{a}^{b}f(x)dx=g(b)-g(a)}

Demo

Sea $<math alttext="{displaystyle {mathcal {P}}={t_{0}<cdots P={t0. .tn!{displaystyle {mathcal {P}}={t_{0} impliedcdots ≤3}{n}}}}}<img alt="{displaystyle {mathcal {P}}={t_{0}<cdots$ partition any of the interval $[a,b]$ , by the theorem of medium value ${displaystyle exists ;x_{i}in [t_{i-1},t_{i}]}$ such as

{displaystyle {begin{aligned}g(t_{i})-g(t_{i-1})&=g'(x_{i})(t_{i}-t_{i-1})\&=f(x_{i})(t_{i}-t_{i-1})end{aligned}}}

Yes

{displaystyle {begin{aligned}m_{i}&=inf{f(x):t_{i-1}leq xleq t_{i}}\M_{i}&=sup{f(x):t_{i-1}leq xleq t_{i}}end{aligned}}}

then

{displaystyle m_{i}(t_{i}-t_{i-1})leq f(x_{i})(t_{i}-t_{i-1})leq M_{i}(t_{i}-t_{i-1})}

that is

{displaystyle m_{i}(t_{i}-t_{i-1})leq g(t_{i})-g(t_{i-1})leq M_{i}(t_{i}-t_{i-1})}

Adding these equations to ${displaystyle i=1,2,dotsn}$ obtained

{displaystyle sum _{i=1}^{n}m_{i}(t_{i}-t_{i-1})leq g(b)-g(a)leq sum _{i=1}^{n}M_{i}(t_{i}-t_{i-1})}

so that

{displaystyle {mathcal {L}}(f,{mathcal {P}})leq g(b)-g(a)leq {mathcal {U}}(f,{mathcal {P}})}

for all partitions $mathcal{P}$ , therefore

{displaystyle int _{a}^{b}{f(t)},dt=g(b)-g(a)}

Examples

Consider the integral

{displaystyle int _{0}^{pi }cos(x)dx}

You have to ${displaystyle F(x)=operatorname {sen}(x)}$ therefore ${displaystyle F'(x)=f(x)=cos(x)}$ for what

{displaystyle {begin{aligned}int _{0}^{pi }cos(x)dx&=operatorname {sen}(x){bigg |}_{0}^{pi }\&=operatorname {sen}(pi)-operatorname {sen}(0)\&=0end{aligned}}}

Consider the integral

{displaystyle int _{1}^{e}{frac {dx}{x}}}

You have to ${displaystyle F(x)=ln(x)}$ therefore ${displaystyle F'(x)=f(x)=1/x}$ for what

{displaystyle {begin{aligned}int _{1}^{e}{frac {dx}{x}}&=ln(x){bigg |}_{1}^{e}\&=ln(e)-ln(1)\&=1end{aligned}}}

As it can be integrated immediately.

Additional bibliography

Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (in English) (2nd edition), New York: John Wiley & Sons, ISBN 978-0-471-00005-1, (requires registration).
Bartle, Robert (2001), A Modern Theory of Integration (in English)AMS, ISBN 0-8218-0845-1.
Leithold, L. (1996), The calculus of a single variable (in English) (6th edition), New York: HarperCollins College Publishers.
Rudin, Walter (1987), Real and Complex Analysis (in English) (third edition), New York: McGraw-Hill Book Co., ISBN 0-07-054234-1.
Courant, Richard; John, Fritz (1965), Introduction to Calculus and Analysis (in English), Springer.
Larson, Ron; Edwards, Bruce H.; Heyd, David E. (2002), Calculus of a single variable (in English) (7th edition), Boston: Houghton Mifflin Company, ISBN 978-0-618-14916-2.
Antoni Malet, Studies on James Gregorie (1638-1675) (PhD Thesis, Princeton, 1989).(in English)
Hernández Rodríguez, O. A.; Lopez Fernández, J. M.. "Teaching the Fundamental Theorem of Calculus: A Historical Reflection", Loci: Convergence (MAA), January 2012. (in English)
Stewart, J. (2003), “Fundamental Theorem of Calculus”, Calculus: early transcendentals (in English), Belmont, California: Thomson/Brooks/Cole.
Turnbull, H. W., ed. (1939), The James Gregory Tercentenary Memorial Volume (in English), London.

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