Integration Formulas and the Net Change Theorem - Revision history

Lila: /* Resources */

2021-10-28T19:17:35Z

Resources

Lila: /* Resources */

2021-10-11T01:52:06Z

Resources

Lila at 01:51, 11 October 2021

2021-10-11T01:51:01Z

Lila: /* Resources */

2021-10-11T01:49:14Z

Resources

Lila: /* Integral of Sine and Cosine */

2021-10-11T01:44:35Z

Integral of Sine and Cosine

Lila: /* Integral of Sine and Cosine */

2021-10-11T01:43:19Z

Integral of Sine and Cosine

Lila: /* Integral of the Exponential function */

2021-10-11T01:42:39Z

Integral of the Exponential function

Lila: /* Integral of the Inverse function */

2021-10-11T01:42:11Z

Integral of the Inverse function

Lila: /* Indefinite integrals of Polynomials */

2021-10-11T01:41:00Z

Indefinite integrals of Polynomials

Lila: /* Basic Properties of Indefinite Integrals */

2021-10-11T01:40:13Z

Basic Properties of Indefinite Integrals

← Older revision		Revision as of 19:17, 28 October 2021
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	* [https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Recognizing_Derivatives_and_the_Substitution_Rule Recognizing Derivatives and Substitution Rule], Wikibooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Recognizing_Derivatives_and_the_Substitution_Rule Recognizing Derivatives and Substitution Rule], Wikibooks: Calculus
	* [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor		* [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral, Wikibooks: Calculus] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Calculus/Area Area, Wikibooks: Calculus] under a CC BY-SA license

@@ Line 131: / Line 131: @@
 ==Resources==
 * [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 * [https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Recognizing_Derivatives_and_the_Substitution_Rule Recognizing Derivatives and Substitution Rule], Wikibooks: Calculus
 * [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor

← Older revision		Revision as of 01:51, 11 October 2021
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	<math>\int x^ndx=\frac{x^{n+1}}{n+1}+C</math>		<math>\int x^ndx=\frac{x^{n+1}}{n+1}+C</math>
		+
		+	==Finding Area or Net Change Between Curves==
		+	Finding the area between two curves, usually given by two explicit functions, is often useful in calculus.
		+
		+	In general the rule for finding the area between two curves is
		+
		+	<math>A=A_{\rm top}-A_{\rm bottom}</math> or
		+
		+	If f(x) is the upper function and g(x) is the lower function
		+
		+	<math>A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx</math>
		+
		+	This is true whether the functions are in the first quadrant or not.
		+
		+	==Area between two curves==
		+	Suppose we are given two functions <math>y_1=f(x)</math> and <math>y_2=g(x)</math> and we want to find the area between them on the interval <math>[a,b]</math> . Also assume that <math>f(x)\ge g(x)</math> for all <math>x</math> on the interval <math>[a,b]</math> . Begin by partitioning the interval <math>[a,b]</math> into <math>n</math> equal subintervals each having a length of <math>\Delta x=\frac{b-a}{n}</math> . Next choose any point in each subinterval, <math>x_i^</math> . Now we can 'create' rectangles on each interval. At the point <math>x_i</math> , the height of each rectangle is <math>f(x_i^)-g(x_i^)</math> and the width is <math>\Delta x</math> . Thus the area of each rectangle is <math>\bigl[f(x_i^)-g(x_i^)\bigr]\Delta x</math> . An ''approximation'' of the area, <math>A</math> , between the two curves is
		+	:<math>A:=\sum_{i=1}^n \Big[f(x_i^)-g(x_i^)\Big]\Delta x</math> .
		+	Now we take the limit as <math>n</math> approaches infinity and get
		+	:<math>A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(x_i^)-g(x_i^)\Big]\Delta x</math>
		+	which gives the exact area. Recalling the definition of the definite integral we notice that
		+	:<math>A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx</math> .
		+	This formula of finding the area between two curves is sometimes known as applying integration with respect to the ''x''-axis since the rectangles used to approximate the area have their bases lying parallel to the ''x''-axis. It will be most useful when the two functions are of the form <math>y_1=f(x)</math> and <math>y_2=g(x)</math> . Sometimes however, one may find it simpler to integrate with respect to the ''y''-axis. This occurs when integrating with respect to the ''x''-axis would result in more than one integral to be evaluated. These functions take the form <math>x_1=f(y)</math> and <math>x_2=g(y)</math> on the interval <math>[c,d]</math> . Note that <math>[c,d]</math> are values of <math>y</math> . The derivation of this case is completely identical. Similar to before, we will assume that <math>f(y)\ge g(y)</math> for all <math>y</math> on <math>[c,d]</math> . Now, as before we can divide the interval into <math>n</math> subintervals and create rectangles to approximate the area between <math>f(y)</math> and <math>g(y)</math> . It may be useful to picture each rectangle having their 'width', <math>\Delta y</math> , parallel to the ''y''-axis and 'height', <math>f(y_i^)-g(y_i^)</math> at the point <math>y_i^*</math>, parallel to the ''x''-axis. Following from the work above we may reason that an ''approximation'' of the area, <math>A</math> , between the two curves is
		+	:<math>A:=\sum_{i=1}^n \Big[f(y_i^)-g(y_i^)\Big]\Delta y</math> .
		+	As before, we take the limit as <math>n</math> approaches infinity to arrive at
		+	:<math>A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(y_i^)-g(y_i^)\Big]\Delta y</math> ,
		+	which is nothing more than a definite integral, so
		+	:<math>A=\int\limits_c^d \bigl(f(y)-g(y)\bigr)dy</math> .
		+	Regardless of the form of the functions, we basically use the same formula.
		+
		+	[[File:Closed path integral defined.png\|Closed_path_integral_defined]]

	==Resources==		==Resources==

@@ Line 101: / Line 101: @@
 ==Resources==
 * [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 * [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor

@@ Line 54: / Line 54: @@
 <math>\int\cos(x)dx=\sin(x)+C</math></br><math>\int\sin(x)dx=-\cos(x)+C</math>
 </blockquote>
 '''Example'''
@@ Line 70: / Line 68: @@
 |<math>=\frac{x^5}{5}+x-2\cos(x)+C</math> .
 |}
 ==Resources==
 * [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 * [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor

@@ Line 51: / Line 51: @@
 So <math>\sin(x)</math> is an antiderivative of <math>\cos(x)</math> and <math>-\cos(x)</math> is an antiderivative of <math>\sin(x)</math> . Hence we get the following rules for integrating <math>\sin(x)</math> and <math>\cos(x)</math>
-{{Calculus/Def|text= <math>\int\cos(x)dx=\sin(x)+C</math></br><math>\int\sin(x)dx=-\cos(x)+C</math>}}
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 We will find how to integrate more complicated trigonometric functions in the chapter on [[Calculus/Integration techniques|integration techniques]].
@@ Line 68: / Line 70: @@
 |<math>=\frac{x^5}{5}+x-2\cos(x)+C</math> .
 |}
 ==Resources==
 * [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 * [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor

@@ Line 39: / Line 39: @@
 :<math>\frac{d}{dx}e^x=e^x</math>
 we see that <math>e^x</math> is its own antiderivative. This allows us to find the integral of an exponential function:
-{{Calculus/Def|text= <math>\int e^xdx=e^x+C</math>}}
 ===Integral of Sine and Cosine===

@@ Line 29: / Line 29: @@
 Therefore, since <math>\frac{1}{x}</math> is the derivative of <math>\ln(x)</math> we can conclude that
-{{Calculus/Def|text= <math>\int\frac{dx}{x}=\ln|x|+C</math>}}
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 Note that the polynomial integration rule does not apply when the exponent is <math>-1</math> . This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.

@@ Line 18: / Line 18: @@
 we have the following rule for indefinite integrals:
-{{Calculus/Def|text= '''Power rule for indefinite integrals'''
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<math>\int x^ndx=\frac{x^{n+1}}{n+1}+C</math> for all <math>n\ne -1</math>}}
+'''Power rule for indefinite integrals'''
 ===Integral of the Inverse function===

@@ Line 1: / Line 1: @@
 ==Indefinite integral identities==
 ===Basic Properties of Indefinite Integrals===
-{{Calculus/Def|text= '''Constant Rule for indefinite integrals'''<br/>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-If <math>c</math> is a constant then <math>\int c\cdot f(x)dx=c\int f(x)dx</math>}}
+'''Constant Rule for indefinite integrals'''
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">