两个多项式函数之间的面积计算
2014-04-19 14:59
本范例演示如何计算两个多项式函数图像之间所夹的面积。系统会自动计算两函数的交点,并利用定积分求出交点之间的面积。同时,通过矩形逼近直观展示积分的几何意义,支持动态调整函数表达式与积分区间,帮助深入理解微积分中面积计算的核心概念。
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="4.4" version="4.4.28.0" id="f97512e6-3017-4bff-83ce-ea1efd8be68a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="785" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.6853503184713375" orientation="0" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="873,258,700,550" />
<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3,3" size="200" window="100,100,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="0" size="538" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="2" size="128" 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" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 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 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="789" height="492"/>
<coordSystem xZero="552.3000000000001" yZero="375.76309346283324" scale="157.8" yscale="21.770874019616024"/>
<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="0" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="789" height="109"/>
<coordSystem xZero="215.0" yZero="315.00000000000006" scale="49.99999999999999" yscale="50.00000000000003"/>
<evSettings axes="false" 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="false" 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"/>
<construction title="" author="" date="">
<expression label ="f" exp="f(x) = (-x^(3)) + (4 * x)"/>
<element type="function" label="f">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<javascript onUpdate=""/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
</element>
<expression label ="g" exp="g(x) = (2 * x^(2)) + x"/>
<element type="function" label="g">
<show object="true" label="false"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<javascript onUpdate=""/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
</element>
<cascell caslabel="rts" >
<cellPair>
<inputCell>
<expression value="rts:=Root[f(x)-g(x)]" eval="Root[f(x) - g(x)]" evalCmd=""/>
</inputCell>
<outputCell>
<expression value="rts:={x = (-3), x = 0, x = 1}" native="true"/>
</outputCell>
</cellPair>
</cascell>
<element type="list" label="rts">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Intersect[Element[rts, k], y = 0]" a1="k" a2="1" a3="Length[rts]"/>
<output a0="rts2"/>
</command>
<element type="list" label="rts2">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="3"/>
</element>
<command name="Sequence">
<input a0="x(Element[rts2, k])" a1="k" a2="1" a3="Length[rts2]"/>
<output a0="roots2"/>
</command>
<element type="list" label="roots2">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Min">
<input a0="roots2"/>
<output a0="xmin"/>
</command>
<element type="numeric" label="xmin">
<value val="-3.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<javascript onUpdate="zoom();"/>
</element>
<command name="Max">
<input a0="roots2"/>
<output a0="xmax"/>
</command>
<element type="numeric" label="xmax">
<value val="1.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<javascript onUpdate="zoom();"/>
</element>
<expression label="xdomain" exp="xmax - xmin" />
<element type="numeric" label="xdomain">
<value val="4.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="padx" exp="xdomain / 8" />
<element type="numeric" label="padx">
<value val="0.5"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="y(Max[f, xmin, xmax])" a1="y(Max[g, xmin, xmax])"/>
<output a0="ymax"/>
</command>
<element type="numeric" label="ymax">
<value val="14.999999226884182"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<javascript onUpdate="zoom();"/>
</element>
<command name="Min">
<input a0="y(Min[f, xmin, xmax])" a1="y(Min[g, xmin, xmax])"/>
<output a0="ymin"/>
</command>
<element type="numeric" label="ymin">
<value val="-3.079201435678004"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<javascript onUpdate="zoom();"/>
</element>
<expression label="yrange" exp="ymax - ymin" />
<element type="numeric" label="yrange">
<value val="18.079200662562187"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="pady" exp="yrange / 8" />
<element type="numeric" label="pady">
<value val="2.2599000828202733"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="Corner1" exp="(xmin - padx, ymin - pady)" type="point" />
<element type="point" label="Corner1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.5" y="-5.3391015184982775" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Corner3" exp="(xmax + padx, ymax + pady)" type="point" />
<element type="point" label="Corner3">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.5" y="17.259899309704455" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Textfield">
<input a0="f"/>
<output a0="inputBox1"/>
</command>
<element type="textfield" label="inputBox1">
<show object="true" label="true" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="11" y="13"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<linkedGeo exp="f"/>
<caption val="f(x)="/>
</element>
<command name="Textfield">
<input a0="g"/>
<output a0="inputBox2"/>
</command>
<element type="textfield" label="inputBox2">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="153" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="6" y="50"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<linkedGeo exp="g"/>
<caption val="g(x)="/>
</element>
<element type="button" label="btnZoom">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<bgColor r="255" g="153" b="204" alpha="255"/>
<layer val="0"/>
<labelOffset x="736" y="22"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<javascript val="zoom();"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<file name="/geogebra/gui/images/nav_skipforward.png"/>
</element>
<expression label="x1" exp="x(Corner1)" />
<element type="numeric" label="x1">
<value val="-3.5"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="y1" exp="y(Corner1)" />
<element type="numeric" label="y1">
<value val="-5.3391015184982775"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="x2" exp="x(Corner3)" />
<element type="numeric" label="x2">
<value val="1.5"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="y2" exp="y(Corner3)" />
<element type="numeric" label="y2">
<value val="17.259899309704455"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Length">
<input a0="roots2"/>
<output a0="nRoots"/>
</command>
<element type="numeric" label="nRoots">
<value val="3.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="Segment[(Element[roots2, k], 0), (Element[roots2, k], f(Element[roots2, k]))]" a1="k" a2="1" a3="nRoots"/>
<output a0="segs"/>
</command>
<element type="list" label="segs">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="IntegralBetween">
<input a0="f" a1="g" a2="xmin" a3="xmax"/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="-10.666666666666666"/>
<show object="true" label="false"/>
<condition showObject="c"/>
<objColor r="255" g="255" b="0" alpha="0.48"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<element type="numeric" label="t">
<value val="0.5"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100.0" x="567.0" y="36.0" fixed="false" horizontal="true"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="n">
<value val="20.0"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="204" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="50" absoluteScreenLocation="true" width="100.0" x="434.0" y="37.0" fixed="false" horizontal="true"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<animation step="1" speed="0.5" type="0" playing="true"/>
</element>
<expression label="dx" exp="xdomain / n" />
<element type="numeric" label="dx">
<value val="0.2"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="xmin + (k * dx)" a1="k" a2="0" a3="n - 1"/>
<output a0="xkLeft"/>
</command>
<element type="list" label="xkLeft">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Element[xkLeft, k] + (t * dx)" a1="k" a2="1" a3="n"/>
<output a0="xk"/>
</command>
<element type="list" label="xk">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="f(Element[xk, k]) - g(Element[xk, k])" a1="k" a2="1" a3="n"/>
<output a0="dyk"/>
</command>
<element type="list" label="dyk">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="f(Element[xk, k])" a1="k" a2="1" a3="n"/>
<output a0="fk"/>
</command>
<element type="list" label="fk">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="g(Element[xk, k])" a1="k" a2="1" a3="n"/>
<output a0="gk"/>
</command>
<element type="list" label="gk">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Polygon[(Element[xkLeft, k], Element[fk, k]), (Element[xkLeft, k] + dx, Element[fk, k]), (Element[xkLeft, k] + dx, Element[gk, k]), (Element[xkLeft, k], Element[gk, k])]" a1="k" a2="1" a3="n"/>
<output a0="rects"/>
</command>
<element type="list" label="rects">
<show object="true" label="false"/>
<condition showObject="b"/>
<objColor r="51" g="51" b="255" alpha="0.5" fillType="1" hatchAngle="45" hatchDistance="5"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="432" y="47"/>
<labelMode val="0"/>
<caption val="分割長條"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="566" y="46"/>
<labelMode val="0"/>
<caption val="函數間面積"/>
</element>
<command name="Sum">
<input a0="rects"/>
<output a0="sumRects"/>
</command>
<element type="numeric" label="sumRects">
<value val="11.869999999999996"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text1" exp=""\sum_{k=1}^{" + (LaTeX[n]) + "} |f(x_k)-g(x_k)| \Delta x=" + (LaTeX[sumRects]) + """ />
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<absoluteScreenLocation x="267" y="14"/>
</element>
</construction>
</geogebra>