函数定积分与黎曼和演示
2014-12-28 13:46
本范例演示了函数定积分的计算及其几何意义。通过GeoGebra指令,直观展示了下和、上和以及定积分的概念。支持多项式函数与三角函数,用户可通过输入框调整积分区间和矩形数量,动态观察黎曼和逼近定积分的过程,是学习微积分基础概念的实用工具。
所属专题
微积分
GeoGebra 画板及绘图内容加载中,请稍等……
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.42.0" id="39bfa0f3-f198-440c-b90e-5fc12d55baba" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.26125" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="869,265,700,550" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="282" window="234,399,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 22 , 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="true" stylebar="true" location="1,1" size="276" window="748,177,692,593" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="789" window="203,439,600,400" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3" size="209" window="865,28,250,638" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="789" height="462"/>
<coordSystem xZero="214.99999999999997" yZero="315.00000000000006" scale="49.999999999999986" yscale="50.00000000000003"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.0" yZero="0.0" zZero="-1.5" scale="50.0" xAngle="20.0" zAngle="-60.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="true"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="true" show="true" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label ="f" exp="f(x) = cos(x) + 2"/>
<element type="function" label="f">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.06" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="5.18" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Min">
<input a0="x(A)" a1="x(B)"/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="0.06"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="x(A)" a1="x(B)"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="5.18"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Integral">
<input a0="f" a1="a" a2="b"/>
<output a0="intF"/>
</command>
<element type="numeric" label="intF">
<value val="9.28738831409232"/>
<show object="false" label="false"/>
<condition showObject="c"/>
<objColor r="102" g="153" b="255" alpha="0.25"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="n">
<value val="9.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="30" absoluteScreenLocation="true" width="100.0" x="606.0" y="403.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="LowerSum">
<input a0="f" a1="a" a2="b" a3="n"/>
<output a0="lowF"/>
</command>
<element type="numeric" label="lowF">
<value val="8.332183369780509"/>
<show object="false" label="false"/>
<condition showObject="e"/>
<objColor r="102" g="0" b="204" alpha="0.5"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label ="g" exp="g(x) = x^(2) / 8 - 2"/>
<element type="function" label="g">
<show object="true" label="false"/>
<condition showObject="i"/>
<objColor r="128" g="0" b="128" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="IntegralBetween">
<input a0="f" a1="g" a2="a" a3="b"/>
<output a0="intFG"/>
</command>
<element type="numeric" label="intFG">
<value val="13.736070980758988"/>
<show object="false" label="false"/>
<condition showObject="d"/>
<objColor r="255" g="0" b="0" alpha="0.44"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<expression label="dx" exp="(b - a) / n" />
<element type="numeric" label="dx">
<value val="0.5688888888888889"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="Polygon[(a + ((k - 1) * dx), g(a + ((k - 0.5) * dx))), (a + (k * dx), g(a + ((k - 0.5) * dx))), (a + (k * dx), f(a + ((k - 0.5) * dx))), (a + ((k - 1) * dx), f(a + ((k - 0.5) * dx)))]" a1="k" a2="1" a3="n"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="true" label="false"/>
<condition showObject="h"/>
<objColor r="75" g="0" b="130" alpha="0.4"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Textfield">
<input a0="g"/>
<output a0="inputBox1"/>
</command>
<element type="textfield" label="inputBox1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="488" y="51"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val=""/>
<linkedGeo exp="g"/>
<caption val="g(x)="/>
</element>
<command name="Textfield">
<input a0="f"/>
<output a0="inputBox2"/>
</command>
<element type="textfield" label="inputBox2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="490" y="15"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val=""/>
<linkedGeo exp="f"/>
<caption val="f(x)="/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="587" y="89"/>
<labelMode val="0"/>
<caption val="∫ f(x) dx"/>
</element>
<element type="boolean" label="d">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="588" y="160"/>
<labelMode val="0"/>
<caption val="∫ (f(x)-g(x)) dx"/>
</element>
<element type="boolean" label="e">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="587" y="123"/>
<labelMode val="0"/>
<caption val="Σ f(x) Δx (lower sum)"/>
</element>
<element type="boolean" label="h">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="588" y="195"/>
<labelMode val="0"/>
<caption val="Σ (f(x)-g(x)) Δx"/>
</element>
<element type="boolean" label="i">
<value val="false"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="443" y="45"/>
<labelMode val="0"/>
<caption val="v"/>
</element>
</construction>
</geogebra>