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<quiz>
  <question type="stack">
    <name>
      <text>27.3.3.2 exercise quadruple integral 1</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>
Evaluate the  triple integral:<br>
\(\displaystyle \int_0^{@dd@}\int_{-1}^{@cc@} \int_{-1}^{@bb@} \int_0^{1-y^2}(x+y^2) \;\mathrm{d}z \,\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}w\ =\ \)[[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p> We are asked to evaluate the  triple integral:<br>
\(\displaystyle \int_0^{@dd@}\int_{-1}^{@cc@} \int_{-1}^{@bb@} \int_0^{1-y^2}(x+y^2) \;\mathrm{d}z \,\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}w \)
<p>We start the integration from the inside and gradually work outwards.</p>

<p>The first integral with respect to \(z\) gives:</p>
<p>\( \displaystyle \int_0^{1-y^2}(x+y^2) \;\mathrm{d}z  = \Bigg[{@fbI11@}\Bigg]_0^{1-y^2} = {@I11@}\) </p>

<p>The second integral, with respect to \(y\) gives: </p>
<p>\( \displaystyle\int_{-1}^{@bb@}  {@I11@}\;\mathrm{d}y =  \Bigg[ {@fbI12@} \Bigg]_{-1}^{@bb@} = {@I12@}\) </p>

<p>The third integral, with respect to \(x\) gives:</p>
<p>\( \displaystyle\int_{-1}^{@cc@} {@I12@}\;\mathrm{d}x = \Bigg[ {@fbI13@} \Bigg]_{-1}^{@bb@} = {@I13@}\)<p>

<p>Finally, integrating with respect to \(w\) gives:</p>
<p>\( \displaystyle\int_0^{@dd@} {@I13@}\;\mathrm{d}w = \Bigg[{@fbI14@}\Bigg]_0^{@dd@} ={@ta1@}\)</p>]]></text>
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    <stackversion>
      <text>2025073100</text>
    </stackversion>
    <questionvariables>
      <text>bb: 1+rand(2)
cc: 1+rand(2)
dd: 1 +rand(2)

fbI11: integrate(x+y^2,z);
I11: integrate(x+y^2,z,0,1-y^2);
fbI12: integrate (I11,y);
I12: integrate (I11,y,-1,bb);
fbI13: integrate (I12,x);
I13: integrate (I12,x,-1,cc);
fbI14: integrate (I13,w);
I14: integrate (I13,w,0,dd);
ta1: I14;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text>[[feedback:prt1]]</text>
    </specificfeedback>
    <questionnote format="html">
      <text>\(\displaystyle \int_0^{@dd@}\int_{-1}^{@cc@} \int_{-1}^{@bb@} \int_0^{1-y^2}(x+y^2) \;\mathrm{d}z \,\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}w = {@ta1@} \)</text>
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    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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      <tans>ta1</tans>
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      <value>1.0000000</value>
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    <qtest>
      <testcase>1</testcase>
      <description>Test case assuming the teacher's input gets full marks.</description>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
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      <testcase>2</testcase>
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      <testinput>
        <name>ans1</name>
        <value>ev(ta1+1,simp)</value>
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