tag:blogger.com,1999:blog-52079419420196078102024-03-14T00:25:19.079-07:00ADD MATHS FROM CD PPSMIsahmozachttp://www.blogger.com/profile/02042923405758991056noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5207941942019607810.post-64261489898514909112009-02-21T00:09:00.000-08:002009-02-21T01:34:24.629-08:00<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='413' height='363' src='https://www.blogger.com/video.g?token=AD6v5dxKAP7xbhhR9k7wROiRFA2tDNJzwI4rob2tTYcUhDc0vxCkKt9Hl28wNURTu1MqHoroT8_M4TxfeKvnRJCykw' class='b-hbp-video b-uploaded' frameborder='0'></iframe>sahmozachttp://www.blogger.com/profile/02042923405758991056noreply@blogger.com0tag:blogger.com,1999:blog-5207941942019607810.post-64529720088124439692009-02-01T01:44:00.000-08:002010-06-06T07:28:55.277-07:00Introduction to Integration<a href="http://www.keepandshare.com/doc/view.php?id=943515&da=y">My module<br /></a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050278&da=y">Introduction 1</a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050282&da=y">Intergartion is revised differentiation 2</a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050283&da=y">Integrsation 3</a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050284&da=y">Animation 4</a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050280&da=y">Aktivity sheet</a><br /><a href="http://www.keepandshare.com/doc/view.php?id=1050281&da=y">Lesson Plan</a>sahmozachttp://www.blogger.com/profile/02042923405758991056noreply@blogger.com1tag:blogger.com,1999:blog-5207941942019607810.post-55052524307798369062009-01-31T06:48:00.000-08:002009-01-31T06:55:03.224-08:00Common Integral<dl compact="compact"><dt>1. </dt><dd> <!-- MATH: $\displaystyle \int adx=ax$ --> <img src="http://www.sosmath.com/tables/integral/integ1/img1.gif" alt="$\displaystyle \int adx=ax$" width="100" align="middle" border="0" height="53" /> <p> </p><p> </p></dd><dt>2. </dt><dd> <!-- MATH: $\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$ --> <img src="http://www.sosmath.com/tables/integral/integ1/img2.gif" alt="$\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$" width="205" align="middle" border="0" height="53" /> <p> </p><p> </p><p> </p></dd><dt>3. </dt><dd> <!-- MATH: $\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx = \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots$ --> <img src="http://www.sosmath.com/tables/integral/integ1/img3.gif" alt="$\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx = \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots $" width="459" align="middle" border="0" height="53" /></dd><dt><br /></dt></dl>sahmozachttp://www.blogger.com/profile/02042923405758991056noreply@blogger.com1tag:blogger.com,1999:blog-5207941942019607810.post-6071051214512490262009-01-31T06:45:00.000-08:002009-01-31T06:48:12.869-08:00What Is Integration<h1 id="firstHeading" class="firstHeading">Integral</h1> <h3 id="siteSub">From Wikipedia, the free encyclopedia</h3> <div id="jump-to-nav">Jump to: <a href="http://en.wikipedia.org/wiki/Integral#column-one">navigation</a>, <a href="http://en.wikipedia.org/wiki/Integral#searchInput">search</a></div> <!-- start content --> <div class="dablink">This article is about the concept of integrals in <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a>. For the set of numbers, see <a href="http://en.wikipedia.org/wiki/Integer" title="Integer">integer</a>. For other uses, see <a href="http://en.wikipedia.org/wiki/Integral_%28disambiguation%29" title="Integral (disambiguation)">Integral (disambiguation)</a>.</div> <table class="infobox" style="margin: 0pt 0pt 1em 1em; width: 180px;" align="right"> <tbody><tr style="background: rgb(204, 204, 255) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" align="center"> <td style="border-bottom: 2px solid rgb(48, 48, 96);"><b>Topics in <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a></b></td> </tr> <tr> <td align="center"> <p><a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a><br /><a href="http://en.wikipedia.org/wiki/Limit_of_a_function" title="Limit of a function">Limits of functions</a><br /><a href="http://en.wikipedia.org/wiki/Continuous_function" title="Continuous function">Continuity</a><br /><a href="http://en.wikipedia.org/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a><br /><a href="http://en.wikipedia.org/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a><br /><a href="http://en.wikipedia.org/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></p> </td> </tr> <tr style="background: rgb(204, 204, 255) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" align="center"> <td><a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative"><b>Differentiation</b></a></td> </tr> <tr> <td align="center"> <p><a href="http://en.wikipedia.org/wiki/Product_rule" title="Product rule">Product rule</a><br /><a href="http://en.wikipedia.org/wiki/Quotient_rule" title="Quotient rule">Quotient rule</a><br /><a href="http://en.wikipedia.org/wiki/Chain_rule" title="Chain rule">Chain rule</a><br /><a href="http://en.wikipedia.org/wiki/Change_of_variables_%28PDE%29" title="Change of variables (PDE)">Change of variables</a><br /><a href="http://en.wikipedia.org/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a><br /><a href="http://en.wikipedia.org/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a><br /><a href="http://en.wikipedia.org/wiki/Related_rates" title="Related rates">Related rates</a><br /><a href="http://en.wikipedia.org/wiki/List_of_differentiation_identities" title="List of differentiation identities">List of differentiation identities</a></p> </td> </tr> <tr style="background: rgb(204, 204, 255) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" align="center"> <td><strong class="selflink"><b>Integration</b></strong></td> </tr> <tr> <td align="center"> <p><a href="http://en.wikipedia.org/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a><br /><a href="http://en.wikipedia.org/wiki/Improper_integral" title="Improper integral">Improper integrals</a><br /><span style="font-family: Times New Roman; font-size: 100%; font-style: italic; font-weight: bold;">Integration by:</span><br /><a href="http://en.wikipedia.org/wiki/Integration_by_parts" title="Integration by parts">parts</a>, <a href="http://en.wikipedia.org/wiki/Disk_integration" title="Disk integration">disks</a>, <a href="http://en.wikipedia.org/wiki/Shell_integration" title="Shell integration">cylindrical<br />shells</a>, <a href="http://en.wikipedia.org/wiki/Integration_by_substitution" title="Integration by substitution">substitution</a>,<br /><a href="http://en.wikipedia.org/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric substitution</a>,<br /><a href="http://en.wikipedia.org/wiki/Partial_fractions_in_integration" title="Partial fractions in integration">partial fractions</a>, <a href="http://en.wikipedia.org/wiki/Order_of_integration_%28calculus%29" title="Order of integration (calculus)">changing order</a></p> </td> </tr> </tbody></table> <table align="right"> <tbody><tr> <td> <div class="thumb tright"> <div class="thumbinner" style="width: 182px;"><a href="http://en.wikipedia.org/wiki/File:Integral_example.png" class="image" title="A definite integral of a function can be represented as the signed area of the region bounded by its graph."><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/42/Integral_example.png/180px-Integral_example.png" class="thumbimage" width="180" border="0" height="180" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Integral_example.png" class="internal" title="Enlarge"><img src="http://en.wikipedia.org/skins-1.5/common/images/magnify-clip.png" alt="" width="15" height="11" /></a></div> A definite integral of a function can be represented as the signed area of the region bounded by its graph.</div> </div> </div> </td> </tr> </tbody></table> <p><b>Integration</b> is an important concept in <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, specifically in the field of <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a> and, more broadly, <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>. Given a <a href="http://en.wikipedia.org/wiki/Function_%28mathematics%29" title="Function (mathematics)">function</a> <i>ƒ</i> of a <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real</a> <a href="http://en.wikipedia.org/wiki/Variable" title="Variable">variable</a> <i>x</i> and an <a href="http://en.wikipedia.org/wiki/Interval_%28mathematics%29" title="Interval (mathematics)">interval</a> [<i>a</i>,<i>b</i>] of the <a href="http://en.wikipedia.org/wiki/Real_line" title="Real line">real line</a>, the <b>integral</b></p> <dl><dd><img class="tex" alt="\int_a^b f(x)\,dx \, ," src="http://upload.wikimedia.org/math/4/0/c/40c4f3fd4e90d666f43f275de6264a1e.png" /></dd></dl> <p>is defined informally to be the signed <a href="http://en.wikipedia.org/wiki/Area_%28geometry%29" title="Area (geometry)" class="mw-redirect">area</a> of the region in the <i>xy</i>-plane bounded by the <a href="http://en.wikipedia.org/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of <i>ƒ</i>, the <i>x</i>-axis, and the vertical lines <i>x</i> = <i>a</i> and <i>x</i> = <i>b</i>.</p> <p>The term "integral" may also refer to the notion of <a href="http://en.wikipedia.org/wiki/Antiderivative" title="Antiderivative">antiderivative</a>, a function <i>F</i> whose <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">derivative</a> is the given function <i>f</i>. In this case it is called an <b>indefinite integral</b>, while the integrals discussed in this article are termed <b>definite integrals</b>. Some authors maintain a distinction between antiderivatives and indefinite integrals.</p> <p>The principles of integration were formulated independently by <a href="http://en.wikipedia.org/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="http://en.wikipedia.org/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibniz</a> in the late seventeenth century. Through the <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, which they independently developed, integration is connected with <a href="http://en.wikipedia.org/wiki/Differential_calculus" title="Differential calculus">differentiation</a>: if <i>f</i> is a continuous real-valued function defined on a <a href="http://en.wikipedia.org/wiki/Closed_interval" title="Closed interval" class="mw-redirect">closed interval</a> [<i>a</i>, <i>b</i>], then, once an antiderivative <i>F</i> of <i>f</i> is known, the definite integral of <i>f</i> over that interval is given by</p> <dl><dd> <dl><dd> <dl><dd><img class="tex" alt="\int_a^b f(x)\,dx = F(b) - F(a)\, ." src="http://upload.wikimedia.org/math/2/3/c/23c4060dcf2db7fe719364df746edcca.png" /></dd></dl> </dd></dl> </dd></dl> Integrals and derivatives became the basic tools of <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a>, with numerous applications in science and <a href="http://en.wikipedia.org/wiki/Engineering" title="Engineering">engineering</a>. A rigorous mathematical definition of the integral was given by <a href="http://en.wikipedia.org/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>. It is based on a <a href="http://en.wikipedia.org/wiki/Limit_%28mathematics%29" title="Limit (mathematics)">limiting</a> procedure which approximates the area of a <a href="http://en.wikipedia.org/wiki/Curvilinear" title="Curvilinear" class="mw-redirect">curvilinear</a> region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integral began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised.sahmozachttp://www.blogger.com/profile/02042923405758991056noreply@blogger.com0