Cavalieri's Principle 3D Demonstration
2018-07-30 16:26
This GeoGebra applet visually demonstrates Cavalieri's Principle. By constructing two solids with the same height and equal cross-sectional areas at every level, it dynamically illustrates that their volumes must be equal. This principle is a fundamental concept in integral calculus for computing volumes, making it ideal for solid geometry and calculus education.
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<input a0="Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ)), (f((i * δ)) + x4, y4, (i * δ))], δ]" a1="i" a2="0" a3="w - 1"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="true" label="false" ev="4"/>
<condition showObject="o1"/>
<objColor r="255" g="204" b="51" alpha="1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ)), (f((i * δ)) + x4, y4, (i * δ))], δ]" a1="i" a2="0" a3="w - 1"/>
<output a0="list2'"/>
</command>
<element type="list" label="list2'">
<show object="true" label="false" ev="4"/>
<condition showObject="o1"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
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<pointStyle val="-1"/>
</element>
<command name="Segment">
<input a0="C" a1="C1"/>
<output a0="f_1"/>
</command>
<element type="segment3d" label="f_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CurveCartesian">
<input a0="f(u) + x4" a1="y4" a2="u" a3="u" a4="0" a5="h"/>
<output a0="u"/>
</command>
<element type="curvecartesian3d" label="u">
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<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
</element>
<command name="Surface">
<input a0="(u * (f(v) + x4)) + ((1 - u) * (f(v) + x3))" a1="(u * y4) + ((1 - u) * y3)" a2="v" a3="u" a4="0" a5="1" a6="v" a7="0" a8="h"/>
<output a0="a"/>
</command>
<element type="surfacecartesian3d" label="a">
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<condition showObject="o2"/>
<objColor r="255" g="204" b="51" alpha="0.5"/>
<layer val="0"/>
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<lineStyle thickness="0" type="0" typeHidden="1"/>
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<command name="Surface">
<input a0="(u * (f(v) + x4)) + ((1 - u) * (f(v) + x1))" a1="(u * y4) + ((1 - u) * y1)" a2="v" a3="u" a4="0" a5="1" a6="v" a7="0" a8="h"/>
<output a0="v"/>
</command>
<element type="surfacecartesian3d" label="v">
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<condition showObject="o2"/>
<objColor r="255" g="204" b="51" alpha="0.5"/>
<layer val="0"/>
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<command name="Polygon">
<input a0="D" a1="E" a2="F" a3="F1"/>
<output a0="q1" a1="d_2" a2="e_2" a3="f_2" a4="f1"/>
</command>
<element type="polygon3d" label="q1">
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<condition showObject="o2"/>
<objColor r="255" g="204" b="51" alpha="0.5"/>
<layer val="0"/>
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<element type="segment3d" label="d_2">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
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<lineStyle thickness="2" type="0" typeHidden="2"/>
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<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="e_2">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
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<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="f_2">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
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</element>
<element type="segment3d" label="f1">
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<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="A" a1="D" a2="F1" a3="C1"/>
<output a0="q2" a1="a_1" a2="d_3" a3="f1_1" a4="c1"/>
</command>
<element type="polygon3d" label="q2">
<lineStyle thickness="2" type="0" typeHidden="2"/>
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="204" b="51" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment3d" label="a_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="d_3">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="f1_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
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<lineStyle thickness="2" type="0" typeHidden="2"/>
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<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="c1">
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<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
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<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="list1'" exp="list1" />
<element type="list" label="list1'">
<show object="true" label="false" ev="4"/>
<condition showObject="o1"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
</construction>
</geogebra>