The Definite Integral - Revision history

Lila: /* Resources */

2021-10-28T16:25:36Z

Resources

Lila at 21:04, 30 September 2021

2021-09-30T21:04:54Z

Lila at 21:01, 30 September 2021

2021-09-30T21:01:44Z

Lila: /* Definition of the Definite Integral */

2021-09-30T20:59:39Z

Definition of the Definite Integral

Lila: /* Linearity with respect to endpoints */

2021-09-30T20:58:27Z

Linearity with respect to endpoints

Lila: /* Linearity with respect to endpoints */

2021-09-30T20:55:21Z

Linearity with respect to endpoints

Lila: /* Basic Properties of the Integral */

2021-09-30T20:54:09Z

Basic Properties of the Integral

Lila: /* Exercises */

2021-09-30T20:49:25Z

Exercises

Lila: /* Examples */

2021-09-30T20:48:37Z

Examples

Lila: /* Introduction */

2021-09-30T20:47:19Z

Introduction

← Older revision		Revision as of 16:25, 28 October 2021
Line 248:		Line 248:

	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Definite%20Integral/MAT1214-5.3TheDefiniteIntegralNotes.pdf The Definite Integral Notes] created by Professor Eduardo Duenez, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/The%20Definite%20Integral/MAT1214-5.3TheDefiniteIntegralNotes.pdf The Definite Integral Notes] created by Professor Eduardo Duenez, UTSA.
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Definite_integral Definite integral, Wikibooks: Calculus] under a CC BY-SA license

@@ Line 91: / Line 91: @@
 :<math>\sum_{i=1}^n a^2=a^2\sum_{i=1}^n 1=na^2</math>
 :<math>\sum_{i=1}^n \frac{2a(b-a)i}{n}=\frac{2a(b-a)}{n}\sum_{i=1}^n i=\frac{2a(b-a)}{n}\cdot\frac{n(n+1)}{2}</math>
-For the third sum we have to use a [[Calculus/Summation_notation|formula]]
+For the third sum we have to use a formula
 :<math>\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}</math>
 to get

@@ Line 139: / Line 139: @@
 There is a special case of this rule used for integrating constants:
-If <math>c</math> is constant then <math>\int\limits_a^b c\ dx=c(b-a)</math>}}
+If <math>c</math> is constant then <math>\int\limits_a^b c\ dx=c(b-a)</math>
 When <math>c>0</math> and <math>a<b</math> this integral is the area of a rectangle of height <math>c</math> and width <math>b-a</math> which equals <math>c(b-a)</math> .
@@ Line 152: / Line 152: @@
 <math>\int\limits_a^b \bigl(f(x)+g(x)\bigr)dx=\int\limits_a^b f(x)dx+\int\limits_a^b g(x)dx</math>
-<math>\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx=\int\limits_a^b f(x)dx-\int\limits_a^b g(x)dx</math>}}
+<math>\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx=\int\limits_a^b f(x)dx-\int\limits_a^b g(x)dx</math>
 As with the constant rule, the addition rule follows from the addition rule for limits:
@@ Line 185: / Line 185: @@
 :<math>\int\limits_a^b f(x)dx\ge\int\limits_a^b g(x)dx</math>
 *Suppose <math>M\ge f(x)\ge m</math> for all <math>x\in[a,b]</math> . Then
-:<math>M(b-a)\ge\int\limits_a^b f(x)dx\ge m(b-a)</math>}}
+:<math>M(b-a)\ge\int\limits_a^b f(x)dx\ge m(b-a)</math>
 If <math>f(x)\ge 0</math> then each of the rectangles in the Riemann sum to calculate the integral of <math>f</math> will be above the <math>y</math> axis, so the area will be non-negative. If <math>f(x)\ge g(x)</math> then <math>f(x)-g(x)\ge 0</math> and by the first property we get the second property. Finally if <math>M\ge f(x)\ge m</math> then the area under the graph of <math>f</math> will be greater than the area of rectangle with height <math>m</math> and less than the area of the rectangle with height <math>M</math> (see [[:File:Integral_comparison_1.png|Figure 7]]).  So
@@ Line 194: / Line 194: @@
 Suppose <math>a<c<b</math> . Then
-:<math>\int\limits_a^b f(x)dx=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx</math>}}
+:<math>\int\limits_a^b f(x)dx=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx</math>
 Again suppose that <math>f</math> is positive. Then this property should be interpreted as saying that the area under the graph of <math>f</math> between <math>a</math> and <math>b</math> is the area between <math>a</math> and <math>c</math> plus the area between <math>c</math> and <math>b</math> (see [[:File:Integral_linear_endpoints.png|Figure 8]]).
@@ Line 206: / Line 206: @@
 With these definitions,
 :<math>\int\limits_a^b f(x)dx=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx</math>
-whatever the order of <math>a,b,c</math> .}}
+whatever the order of <math>a,b,c</math> .
 ===Even and odd functions===
@@ Line 217: / Line 217: @@
 If <math>f</math> is a continuous even function then for any <math>a</math>,
-:<math>\int\limits_{-a}^a f(x)dx=2\int\limits_0^a f(x)dx</math>}}
+:<math>\int\limits_{-a}^a f(x)dx=2\int\limits_0^a f(x)dx</math>
 Suppose <math>f</math> is an odd function and consider first just the integral from <math>-a</math> to <math>0</math> . We make the substitution <math>u=-x</math> so <math>du=-dx</math> . Notice that if <math>x=-a</math> then <math>u=a</math> and if <math>x=0</math> then <math>u=0</math> . Hence
@@ Line 239: / Line 239: @@
 :<math>\int\limits_{-a}^0 f(x)dx=\int\limits_a^0 f(-u)(-du)=-\int\limits_a^0 f(-u)du=\int\limits_0^a f(-u)du=\int\limits_0^a f(u)du</math>
 where the last step has used the evenness of <math>f</math> . Since <math>u</math> is just a dummy variable, we can replace it with <math>x</math> . Then
-:<math>\int\limits_{-a}^a f(x)dx=\int\limits_0^a f(x)dx+\int\limits_0^a f(x)dx=2\int\limits_0^a f(x)dx</math>{{noprint/bottom}}}}
+:<math>\int\limits_{-a}^a f(x)dx=\int\limits_0^a f(x)dx+\int\limits_0^a f(x)dx=2\int\limits_0^a f(x)dx</math>
 ==Resources==

@@ Line 22: / Line 22: @@
 Suppose <math>f</math> is a continuous function on <math>[a,b]</math> and <math>\Delta x=\frac{b-a}{n}</math> . Then the ''definite integral'' of <math>f</math> between <math>a</math> and <math>b</math> is
 :<math>\int\limits_a^b f(x)dx=\lim_{n\to\infty}A_n=\lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x</math>
-where <math>x_i^*</math> are any sample points in the interval <math>[x_{i-1},x_i]</math> and <math>x_k=a+k\cdot\Delta x</math> for <math>k=0,\dots,n</math> .}}
+where <math>x_i^*</math> are any sample points in the interval <math>[x_{i-1},x_i]</math> and <math>x_k=a+k\cdot\Delta x</math> for <math>k=0,\dots,n</math> .
 It is a fact that if <math>f</math> is continuous on <math>[a,b]</math> then this limit always exists and does not depend on the choice of the points <math>x_i^*\in[x_{i-1},x_i]</math> . For instance they may be evenly spaced, or distributed ambiguously throughout the interval. The proof of this is technical and is beyond the scope of this section.
 ===Notation===
-When considering the expression, <math>\int\limits_a^b f(x)dx</math> (read "the integral from <math>a</math> to <math>b</math> of the <math>f</math> of <math>x</math> <math>dx</math>"), the function <math>f</math> is called the ''integrand'' and the interval <math>[a,b]</math> is the interval of integration. Also <math>a</math> is called the ''lower limit'' and <math>b</math> the ''upper limit'' of integration.}}
+When considering the expression, <math>\int\limits_a^b f(x)dx</math> (read "the integral from <math>a</math> to <math>b</math> of the <math>f</math> of <math>x</math> <math>dx</math>"), the function <math>f</math> is called the ''integrand'' and the interval <math>[a,b]</math> is the interval of integration. Also <math>a</math> is called the ''lower limit'' and <math>b</math> the ''upper limit'' of integration.
 [[Image:signed-area.png|thumb|right|300px|Figure 4: The integral gives the signed area under the graph.]]

@@ Line 199: / Line 199: @@
 [[Image:Integral_linear_endpoints.png|thumb|300px|right|Figure 8:  Illustration of the property of additivity with respect to endpoints]]
-Extension of Additivity with respect to limits of integration:
+{{Calculus/Def|text= '''Extension of Additivity with respect to limits of integration'''<br>
 When <math>a=b</math> we have that <math>\Delta x=\frac{b-a}{n}=0</math> so
 :<math>\int\limits_a^a f(x)dx=0</math>

@@ Line 117: / Line 117: @@
 ===The Constant Rule===
-{{Calculus/Def|text= '''Constant Rule'''
+Constant Rule: <math>\int\limits_a^b c\cdot f(x)dx=c\int\limits_a^b f(x)dx</math>
 When <math>f</math>  is positive, the height of the function <math>c\cdot f</math> at a point <math>x</math> is <math>c</math> times the height of the function <math>f</math> .  So the area under <math>c\cdot f</math> between <math>a</math> and <math>b</math> is <math>c</math> times the area under <math>f</math> . We can also give a proof using the definition of the integral, using the constant rule for limits,
@@ Line 140: / Line 139: @@
 There is a special case of this rule used for integrating constants:
 If <math>c</math> is constant then <math>\int\limits_a^b c\ dx=c(b-a)</math>}}
@@ Line 151: / Line 149: @@
 ===The addition and subtraction rule===
-{{Calculus/Def|text= '''Addition and Subtraction Rules of Integration'''<br>
+Addition and Subtraction Rules of Integration:
 <math>\int\limits_a^b \bigl(f(x)+g(x)\bigr)dx=\int\limits_a^b f(x)dx+\int\limits_a^b g(x)dx</math>
 <math>\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx=\int\limits_a^b f(x)dx-\int\limits_a^b g(x)dx</math>}}
@@ Line 176: / Line 175: @@
 '''Example'''
 :<math>\int\limits_0^2 (4x^2+14)dx=4\int\limits_0^2 x^2dx+\int\limits_0^2 14dx=4\cdot\frac{1}{3}(2^3-0^3)+2\cdot 14=\frac{32}{3}+28=\frac{116}{3}</math>
 ===The Comparison Rule===
 [[Image:Integral_comparison_1.png|thumb|300px|right|Figure 7: Bounding the area under <math>f(x)</math> on <math>[a,b]</math>]]
-{{Calculus/Def|text= '''Comparison Rule'''
+Comparison Rule:
 *Suppose <math>f(x)\ge 0</math> for all <math>x\in[a,b]</math> . Then
 :<math>\int\limits_a^b f(x)dx\ge 0</math>
@@ Line 196: / Line 191: @@
 ===Linearity with respect to endpoints===
-{{Calculus/Def|text= '''Additivity with respect to endpoints'''
+Additivity with respect to endpoints:
 Suppose <math>a<c<b</math> . Then
 :<math>\int\limits_a^b f(x)dx=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx</math>}}
@@ Line 211: / Line 207: @@
 :<math>\int\limits_a^b f(x)dx=\int\limits_a^c f(x)dx+\int\limits_c^b f(x)dx</math>
 whatever the order of <math>a,b,c</math> .}}
 ===Even and odd functions===
 Recall that a function <math>f</math> is called odd if it satisfies <math>f(-x)=-f(x)</math> and is called even if <math>f(-x)=f(x)</math> .
-{{Calculus/Def|text= Suppose <math>f</math> is a continuous odd function then for any <math>a</math> ,
+Suppose <math>f</math> is a continuous odd function. Then for any <math>a</math>,
 :<math>\int\limits_{-a}^a f(x)dx=0</math>
-If <math>f</math> is a continuous even function then for any <math>a</math> ,
+If <math>f</math> is a continuous even function then for any <math>a</math>,
 :<math>\int\limits_{-a}^a f(x)dx=2\int\limits_0^a f(x)dx</math>}}
@@ Line 237: / Line 231: @@
 The proof of the formula for even functions is similar.
-{{question-answer|question=5. Prove that if <math>f</math> is a continuous even function then for any <math>a</math> ,
+Prove that if <math>f</math> is a continuous even function then for any <math>a</math> ,
-:<math>\int\limits_{-a}^a f(x)dx=2\int\limits_0^a f(x)dx</math> .|answer={{noprint/top}}From the property of linearity of the endpoints we have<br/>
+:<math>\int\limits_{-a}^a f(x)dx=2\int\limits_0^a f(x)dx</math>.
 :<math>\int\limits_{-a}^a f(x)dx=\int\limits_{-a}^0 f(x)dx+\int\limits_0^a f(x)dx</math>
 Make the substitution <math>u=-x;\quad du=-dx</math> . <math>u=a</math> when <math>x=-a</math> and <math>u=0</math> when <math>x=0</math> . Then

@@ Line 112: / Line 112: @@
 |<math>=\frac{1}{3}(b^3-a^3)</math>
 |}
 ==Basic Properties of the Integral==

@@ Line 61: / Line 61: @@
 Notice that <math>f(x_i^*)=x_i^*=\frac{i}{n}</math> . Putting this into the formula for the approximation,
 :<math>A_n=\sum_{i=1}^n f(x_{i}^*)\Delta x=\sum_{i=1}^n f\left(\frac{i}{n}\right)\Delta x=\sum_{i=1}^n \frac{i}{n}\cdot\frac{1}{n}=\frac{1}{n^2}\sum_{i=1}^n i</math>
-Now we use the [[Calculus/Summation_notation|formula]]
+Now we use the formula
 :<math>\sum_{i=1}^n i=\frac{n(n+1)}{2}</math>
 to get

@@ Line 15: / Line 15: @@
 :<math>A_n=f(x_1^*)\Delta x+\dots+f(x_n^*)\Delta x</math>
-A more convenient way to write this is with [[Calculus/Summation notation| summation notation]]:
+A more convenient way to write this is with summation notation.
 :<math>A_n=\sum_{i=1}^n f(x_i^*)\Delta x</math>
 For each number <math>n</math> we get a different approximation. As <math>n</math> gets larger the width of the rectangles gets smaller which yields a better approximation (see [[:File:Riemann2.gif|Figure 3]]). In the limit of <math>A_n</math> as <math>n</math> tends to infinity we get the area <math>S</math> .
 ==Definition of the Definite Integral==