3D Solid Construction with Square Cross-Sections over Two Functions
2014-12-29 10:49
This applet demonstrates constructing a 3D solid with square cross-sections over two function curves. Using Surface, Prism, and Curve commands, it visualizes finding volumes by known cross-sectional areas in calculus. Users can observe the dynamic relationship between function graphs and solid cross-sections, deepening their understanding of definite integrals for calculating volumes.
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<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="Midpoint[P_{dx}, P_{flag}]"/>
<labelOffset x="8" y="4"/>
</element>
<expression label="k" exp="x = x(J)" type="plane" />
<element type="plane3d" label="k">
<show object="true" label="false"/>
<condition showObject="showPlane"/>
<objColor r="255" g="215" b="0" alpha="0.35"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.0" y="0.0" z="0.0" w="-1.2274481561559847"/>
<fading val="0.0"/>
</element>
<command name="CurveCartesian">
<input a0="t" a1="f(t)" a2="abs(f(t) - g(t))" a3="t" a4="x1" a5="x2"/>
<output a0="m"/>
</command>
<element type="curvecartesian3d" label="m">
<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="0" typeHidden="1"/>
</element>
<command name="CurveCartesian">
<input a0="t" a1="g(t)" a2="abs(f(t) - g(t))" a3="t" a4="x1" a5="x2"/>
<output a0="n"/>
</command>
<element type="curvecartesian3d" label="n">
<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="0" typeHidden="1"/>
</element>
<element type="plane3d" label="o">
<show object="true" label="false"/>
<objColor r="160" g="160" b="160" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<fading val="0.0"/>
</element>
<command name="Point">
<input a0="segFlag"/>
<output a0="P_{section}"/>
</command>
<element type="point3d" label="P_{section}">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<ggbscript val="SetValue[showPlane, !showPlane]"/>
<coords x="-2.4827932574436957" y="2.0507276787941713" z="2.270032333908836" w="1.0"/>
<pointSize val="4"/>
</element>
<expression label="txtShowPlane" exp=""Section Plane""/>
<element type="text" label="txtShowPlane">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<startPoint exp="P_{section}"/>
<labelOffset x="6" y="19"/>
<ggbscript val="RunClickScript[P_{section}]"/>
</element>
<element type="boolean" label="showPlane">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
</construction>
</geogebra>