Simpson's rule - Revision history

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T19:20:37Z

‎top: clean up & uniformity

2009-11-29T18:57:29Z

Reverted edits by SmytheQ (Talk) to last version by WilliamBeason

2009-11-29T18:52:51Z

2008-12-31T16:26:36Z

added link and reworded sentence

2008-12-31T15:40:11Z

‎Simpson's Rule: added a picture for example 2

2008-12-31T15:16:09Z

‎Example 1: added further explanations

2008-12-31T15:11:46Z

‎Example 2 Continued: rewording to improve flow

2008-12-31T15:09:49Z

‎Example 2 Continued: misc text addition

2008-12-31T15:08:27Z

‎Example 2 Continued: reformatting and changing text order

2008-12-31T15:06:17Z

‎Example 1: added text above example pic

@@ Line 1: / Line 1: @@
-{{Template:Math-h}}
+{{Math-h}}
-In [[Calculus]], Simpson's Rule is a method of [[estimate|estimating]] integrals by using [[parabolas]] or other higher order [[polynomial]] expressions. It is named after its discoverer, [[Thomas Simpson]]<ref>http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html</ref>.
+In [[Calculus]], Simpson's Rule is a method of [[estimate|estimating]] integrals by using [[parabolas]] or other higher order [[polynomial]] expressions. It is named after its discoverer, [[Thomas Simpson]].<ref>http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html</ref>
 ==Simpson's Rule==

@@ Line 46: / Line 46: @@
 :<math>a = 1</math>
 :<math>b = 4</math>
 :<math>f(x)=-x^3 + 4x^2-x+1</math>
 :<math>f(1)=-1^3 + 4(1)^2 - 1 + 1 = 3</math>

@@ Line 46: / Line 46: @@
 :<math>a = 1</math>
 :<math>b = 4</math>
 :<math>f(x)=-x^3 + 4x^2-x+1</math>
 :<math>f(1)=-1^3 + 4(1)^2 - 1 + 1 = 3</math>

@@ Line 1: / Line 1: @@
 {{Template:Math-h}}
-In [[Calculus]], Simpson's Rule is a method of approximating integrals by using [[parabolas]] or other higher order [[polynomial]] expressions. It is named after its discoverer, [[Thomas Simpson]]<ref>http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html</ref>.
+In [[Calculus]], Simpson's Rule is a method of [[estimate|estimating]] integrals by using [[parabolas]] or other higher order [[polynomial]] expressions. It is named after its discoverer, [[Thomas Simpson]]<ref>http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html</ref>.
 ==Simpson's Rule==

@@ Line 57: / Line 57: @@
 === Example 2 ===
 Given:
 : <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>
@@ Line 62: / Line 64: @@
 :<math>a = 4</math>
 :<math>b = 12</math>
 :<math>f(x)=\frac{3x+11}{x^2-x-6}</math>
 :<math>f(4) = \frac{3(4)+11}{4^2-4-6} = {23 \over 6}</math>

@@ Line 52: / Line 52: @@
 <br />Now we can evaluate the integral:
 :<math>{{4-1} \over 6} \left [ 3 + 4(7.875) - 3 \right ] = \left ({1 \over 2} \right)(31.5) = 15.75</math>
 If we solve this using [[integration]], we find that Simpson's Rule gave the '''exact''' value for the definite integral. ''However, this is rarely the case'', as shown in the next example.

@@ Line 108: / Line 108: @@
 :<math>=\left ( { 1 \over 3 } \right ) \left( {68579\over2860} \right) = {68579\over8580}</math>
 :<math>\approx 7.99289...</math>
-<br />This new answer is much closer than the original; it is off by about <math>.05</math>. Note how much less "error" area that there is on the picture. The graph represents the first parabola we use for Example 2 Continued, evaluating from 4 to 6. By reducing the area that each individual parabola approximates, our total approximation became more accurate.
+<br />This new answer is much closer than the original; it is off by about <math>.05</math>. In the picture on the right, note how much less "error" area that there is compared to when we only used one parabola. The graph represents the first parabola we use for Example 2 Continued, evaluating from 4 to 6. By reducing the interval that each individual parabola approximates, our total approximation became more accurate.
   If the number of subintervals is increased, it can be shown that the estimated value approaches the real value:

@@ Line 94: / Line 94: @@
 |-
 |[[Image:Simpsonsrule2cd7.png|250px|Click on the image to get a better view.]]
 |-
 |}

@@ Line 11: / Line 11: @@
 {| class="wikitable" style="float: right"
 |-
-| colspan="2" align="center"| [[Image:Simpsonsruleat9.png|300px|Click on the image to get a better view.]]
+|colspan="2" align="center"| [[Image:Simpsonsruleat9.png|300px|Click on the image to get a better view.]]
 |-
 !width="200"|Explanation