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Difference between revisions of "Integral"

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(→‎Definite Integral)
 
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The '''integral''' is a generalization of [[area]].  The integral of a [[function]] is defined as the area between it and the <math>x</math>-axis.  If the function lies below the <math>x</math>-axis, then the area is negative.
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The '''integral''' is one of the two base concepts of [[calculus]], along with the [[derivative]].
  
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==Basic integrals==
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==Beginner Level==
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In introductory, high-school level texts, the integral is often presented in two parts, the '''indefinite integral''' and '''definite integral'''. Although this approach lacks mathematical formality, it has the advantage of being easy to grasp and convenient to use in most of its applications, especially in [[Physics]].
  
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==Properties of integrals==
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===Indefinite Integral===
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<math>\int_{a}^b f = \int_a^c f + \int_c^b f</math>
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The indefinite integral, or antiderivative, is a partial [[inverse]] of the [[derivative]]. That is, if the  derivative of a [[function ]]<math>f(x)</math> is written as <math>f'(x)</math>, then the indefinite integral of <math>f'(x)</math> is <math>f(x)+c</math>, where <math>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>.
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====Notation====
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*The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where the <math>dx</math> means that the function is being integrated in relation to <math>x</math>.
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*Often, to save space, the integral of <math>f(x)</math> is written as <math>F(x)</math>, the integral of <math>h(x)</math> as <math>H(x)</math>, etc.
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====Rules of Indefinite Integrals====
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*<math>\int c\,dx=cx+C</math> for a constant <math>c</math> and another constant <math>C</math>.
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*<math>\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx</math>
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*<math>\int x^n\,dx=\frac{1}{n+1}x^{n+1}+C</math>, <math>n \ne -1</math>
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*<math>\int x^{-1}\,dx=\ln |x|+C</math>
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*<math>\int \sin x\,dx = -\cos x + C</math>
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*<math>\int \cos x\,dx = \sin x + C</math>
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*<math>\int\tan x\,dx =  \ln |\cos x| + C</math>
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*<math>\int \sec x\,dx = \ln |\sec x + \tan x| + C</math>
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*<math>\int \csc \, dx =\ln |\csc x + \cot x| + C</math>
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*<math>\int \cot x\,dx = \ln |\sin x| + C</math>
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*<math>\int cf(x)\, dx=c\int f(x)\,dx</math>
  
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===Definite Integral===
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The definite integral is also the [[area]] under a [[curve]] between two [[points]] <math>a</math> and <math>b</math>. For example, the area under the curve <math>f(x)=\sin x</math> between <math>-\frac{\pi}{2}</math> and <math>\frac{\pi}{2}</math> is <math>0</math>, as area below the x-axis is taken as negative area.
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====Definition and Notation====
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*The definite integral of a function between <math>a</math> and <math>b</math> is written as <math>\int^{b}_{a}f(x)\,dx</math>.
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*<math>\int^{b}_{a}f(x)\,dx=F(b)-F(a)</math>, where <math>F(x)</math> is the antiderivative of <math>f(x)</math>. This is also notated <math>[\int f(x)\,dx]^{b}_{a}</math>, read as "The integral of <math>f(x)</math> evaluated at <math>a</math> and <math>b</math>." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
  
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==Other uses==
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====Rules of Definite Integrals====
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The word ''integral'' is the adjectival form of the noun "[[integer]]."  Thus, <math>3</math> is integral while <math>\pi</math> is not.
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*<math>\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}f(x)\,dx</math> for any <math>c</math>.
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Also,
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<cmath>\text{Area} = \iint f(x, y)\,dA</cmath>,
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<cmath>\text{Vol} = \iiint f(x, y, z)\,dV</cmath>,
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<cmath>\text{4V} = \iiiint f(x, y, z, w)\,dV</cmath>.
  
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The word ''integral'' is also used in English to describe the state of being integrated; e.g., "The engine is an integral part of the vehicle -- without it, nothing would work."
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==Formal Use==
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The notion of an '''integral''' is one of the key ideas in severel areas of higher mathematics including [[analysis]] and [[topology]]. The integral can be defined in several ways which can be applied to several different settings. However, the most common definition, and the one which most closely resembles the the 'definite integral' is the '''Riemann Integral'''
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===Riemann Integral===
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Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
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Let <math>L\in\mathbb{R}</math>
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We say that <math>f</math> is '''Riemann Integrable''' on <math>[a,b]</math> if and only if
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<math>\forall \epsilon>0\:\exists\delta>0</math> such that if <math>\mathcal{\dot{P}}</math> is a [[Partition|tagged partition]] on <math>[a,b]</math> with <math>\|\mathcal{\dot{P}}\|<\delta</math> <math>\implies</math> <math>|S(f,\mathcal{\dot{P}})-L|<\epsilon</math>, where <math>S(f,\mathcal{\dot{P}})</math> is the [[Riemann sum]] of <math>f</math> with respect to <math>\mathcal{\dot{P}}</math>
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<math>L</math> is said to be the '''integral''' of <math>f</math> on <math>[a,b]</math> and is written as <math>L=\int_a^b f(x)dx=\int_a^b f</math>
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<geogebra>2f3876024e3b8d9e4506f2173c591cbfaca665de</geogebra>
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Another integral commonly used in introductory texts is the '''Darboux Integral''' (which is often called the Riemann Integral)
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===Darboux Integral===
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Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
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We say that <math>f</math> is '''Darboux Integrable''' on <math>[a,b]</math> if and only if <math>\inf_{\mathcal{P}}U(f,\mathcal{P})=\sup_{\mathcal{P}}L(f,\mathcal{P})</math>, where <math>L(f,\mathcal{P})</math> and <math>U(f,\mathcal{P})</math> are respectively the [[Riemann sum|lower sum]] and [[Riemann sum|upper sum]] of <math>f</math> with respect to [[partition]] <math>\mathcal{P}</math>
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The notation used for the Darboux integral is the same as that for the Riemann integral.
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{{image}}
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===Other Definitions===
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Other important definitions of integration include the [[Riemann-Stieltjes integral]], [[Lebesgue integral]], [[Henstock-Kurzweil integral]], etc.
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==Disambiguation==
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*The word ''integral'' is the adjectival form of the noun "[[integer]]."  Thus, <math>3</math> is integral while <math>\pi</math> is not.
  
 
==See also==
 
==See also==
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*[[Derivative]]
 
*[[Derivative]]
 
*[[Function]]
 
*[[Function]]
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*[[Chain Rule]]
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*[[Measure theory]]
  
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{{stub}}
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[[Category:Calculus]]
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[[Category:Definition]]

Latest revision as of 15:38, 13 November 2024

The integral is one of the two base concepts of calculus, along with the derivative.

Beginner Level

In introductory, high-school level texts, the integral is often presented in two parts, the indefinite integral and definite integral. Although this approach lacks mathematical formality, it has the advantage of being easy to grasp and convenient to use in most of its applications, especially in Physics.

Indefinite Integral

The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function $f(x)$ is written as $f'(x)$, then the indefinite integral of $f'(x)$ is $f(x)+c$, where $c$ is a real constant. This is because the derivative of a constant is $0$.

Notation

  • The integral of a function $f(x)$ is written as $\int f(x)\,dx$, where the $dx$ means that the function is being integrated in relation to $x$.
  • Often, to save space, the integral of $f(x)$ is written as $F(x)$, the integral of $h(x)$ as $H(x)$, etc.

Rules of Indefinite Integrals

  • $\int c\,dx=cx+C$ for a constant $c$ and another constant $C$.
  • $\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx$
  • $\int x^n\,dx=\frac{1}{n+1}x^{n+1}+C$, $n \ne -1$
  • $\int x^{-1}\,dx=\ln |x|+C$
  • $\int \sin x\,dx = -\cos x + C$
  • $\int \cos x\,dx = \sin x + C$
  • $\int\tan x\,dx = \ln |\cos x| + C$
  • $\int \sec x\,dx = \ln |\sec x + \tan x| + C$
  • $\int \csc \, dx =\ln |\csc x + \cot x| + C$
  • $\int \cot x\,dx = \ln |\sin x| + C$
  • $\int cf(x)\, dx=c\int f(x)\,dx$

Definite Integral

The definite integral is also the area under a curve between two points $a$ and $b$. For example, the area under the curve $f(x)=\sin x$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ is $0$, as area below the x-axis is taken as negative area.

Definition and Notation

  • The definite integral of a function between $a$ and $b$ is written as $\int^{b}_{a}f(x)\,dx$.
  • $\int^{b}_{a}f(x)\,dx=F(b)-F(a)$, where $F(x)$ is the antiderivative of $f(x)$. This is also notated $[\int f(x)\,dx]^{b}_{a}$, read as "The integral of $f(x)$ evaluated at $a$ and $b$." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.

Rules of Definite Integrals

  • $\int^{b}_{a}f(x)\,dx=\int^{b}_{c}f(x)\,dx+\int^{c}_{a}f(x)\,dx$ for any $c$.

Also, \[\text{Area} = \iint f(x, y)\,dA\], \[\text{Vol} = \iiint f(x, y, z)\,dV\], \[\text{4V} = \iiiint f(x, y, z, w)\,dV\].

Formal Use

The notion of an integral is one of the key ideas in severel areas of higher mathematics including analysis and topology. The integral can be defined in several ways which can be applied to several different settings. However, the most common definition, and the one which most closely resembles the the 'definite integral' is the Riemann Integral

Riemann Integral

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $L\in\mathbb{R}$

We say that $f$ is Riemann Integrable on $[a,b]$ if and only if

$\forall \epsilon>0\:\exists\delta>0$ such that if $\mathcal{\dot{P}}$ is a tagged partition on $[a,b]$ with $\|\mathcal{\dot{P}}\|<\delta$ $\implies$ $|S(f,\mathcal{\dot{P}})-L|<\epsilon$, where $S(f,\mathcal{\dot{P}})$ is the Riemann sum of $f$ with respect to $\mathcal{\dot{P}}$

$L$ is said to be the integral of $f$ on $[a,b]$ and is written as $L=\int_a^b f(x)dx=\int_a^b f$

<geogebra>2f3876024e3b8d9e4506f2173c591cbfaca665de</geogebra>

Another integral commonly used in introductory texts is the Darboux Integral (which is often called the Riemann Integral)

Darboux Integral

Let $f:[a,b]\rightarrow\mathbb{R}$

We say that $f$ is Darboux Integrable on $[a,b]$ if and only if $\inf_{\mathcal{P}}U(f,\mathcal{P})=\sup_{\mathcal{P}}L(f,\mathcal{P})$, where $L(f,\mathcal{P})$ and $U(f,\mathcal{P})$ are respectively the lower sum and upper sum of $f$ with respect to partition $\mathcal{P}$

The notation used for the Darboux integral is the same as that for the Riemann integral.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

Other Definitions

Other important definitions of integration include the Riemann-Stieltjes integral, Lebesgue integral, Henstock-Kurzweil integral, etc.

Disambiguation

  • The word integral is the adjectival form of the noun "integer." Thus, $3$ is integral while $\pi$ is not.

See also

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