垫圈法计算中空旋转体体积
2015-01-02 14:21
本范例演示如何使用垫圈法（Washer Method）计算中空旋转体的体积。通过积分公式 π∫[a,b]([f(x)]²-[g(x)]²)dx，直观展示由两条曲线 f(x) 和 g(x) 绕轴旋转所形成的立体图形。用户可以调整函数和积分区间，观察旋转体截面（垫圈）的变化及体积的计算过程，帮助深入理解定积分在立体几何中的应用。
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<command name="CircleSector">
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<output a0="thetaSector"/>
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<element type="conicpart" label="thetaSector">
<show object="true" label="false" ev="3"/>
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<expression label="txtShowDisks" exp=""Show Washers""/>
<element type="text" label="txtShowDisks">
<show object="true" label="false"/>
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<element type="numeric" label="maxDisks">
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<show object="false" label="true"/>
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<layer val="2"/>
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<animation step="0.1" speed="1" type="0" playing="false"/>
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<command name="Point">
<input a0="Segment[P32, P3]"/>
<output a0="P4"/>
</command>
<element type="point3d" label="P4">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
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<expression label="txtIntegral" exp=""V=\pi\int_{a}^{b}{ [f(x)]^2}dx""/>
<element type="text" label="txtIntegral">
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<command name="Segment">
<input a0="A" a1="(x(A), f(x(A)))"/>
<output a0="segA"/>
</command>
<element type="segment" label="segA">
<show object="true" label="false"/>
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<command name="Segment">
<input a0="B" a1="(x(B), f(x(B)))"/>
<output a0="segB"/>
</command>
<element type="segment" label="segB">
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<expression label="text1" exp=""f(x)=" + (LaTeX[f]) + """ />
<element type="text" label="text1">
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<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
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<command name="CurveCartesian">
<input a0="t" a1="(g(t) * cos(θ1))" a2="(g(t) * sin(θ1))" a3="t" a4="x1" a5="x2"/>
<output a0="gCurveT1"/>
</command>
<element type="curvecartesian3d" label="gCurveT1">
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<command name="CircleArc">
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<output a0="thetaArc"/>
</command>
<element type="conicpart" label="thetaArc">
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<command name="CurveCartesian">
<input a0="t" a1="(g(t) * cos(θ2))" a2="(g(t) * sin(θ2))" a3="t" a4="x1" a5="x2"/>
<output a0="gCurveT2"/>
</command>
<element type="curvecartesian3d" label="gCurveT2">
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<command name="Surface">
<input a0="t" a1="(g(t) * cos(s))" a2="(g(t) * sin(s))" a3="t" a4="a" a5="b" a6="s" a7="θ1" a8="θ2"/>
<output a0="surfaceG"/>
</command>
<element type="surfacecartesian3d" label="surfaceG">
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<levelOfDetail val="5"/>
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<command name="Sequence">
<input a0="QuadricSide[Cylinder[(x1 + ((k - 1) * dx), 0, 0), (x1 + (k * dx), 0, 0), f(x1 + ((k - 1 / 2) * dx))]]" a1="k" a2="1" a3="n"/>
<output a0="sidesF"/>
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<element type="list" label="sidesF">
<show object="true" label="false"/>
<condition showObject="showWashers"/>
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<command name="Sequence">
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<output a0="ringsL"/>
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<element type="list" label="ringsL">
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<condition showObject="showWashers"/>
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<expression label="f2" exp="f2(x) = If[x1 ≤ x ≤ x2,f]" />
<element type="function" label="f2">
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<expression label="g2" exp="g2(x) = If[x1 ≤ x ≤ x2,g]" />
<element type="function" label="g2">
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<expression label="text2" exp=""g(x)=" + (LaTeX[g]) + """ />
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<command name="Textfield">
<input a0="g"/>
<output a0="inputBox2"/>
</command>
<element type="textfield" label="inputBox2">
<show object="true" label="true" ev="3"/>
<objColor r="128" g="0" b="128" alpha="1.0"/>
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<labelOffset x="58" y="215"/>
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<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<linkedGeo exp="g"/>
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<caption val="g(x)="/>
</element>
<expression label="n2" exp="ceil((y(P4 - P32) / y(P3 - P32) * maxDisks))" />
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<expression label="txtNumDisks" exp=""↕ Washers：" + n + """ />
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<command name="IntegralBetween">
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</construction>
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