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12.7 - Triple Integrals / Volumes
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[12.7] - Triple integrals / Volume 
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<li>Find the volume of the figures shown below, using both a triple integral, and geometry.<br>
<img src="0g/12.7.fig1.jpg" style="width:250px"><br>


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<img src="0g/12.7.fig2.jpg" style="width:250px"><br>

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Sketch the solid whose volume is given by
$$\int_0^4\int_0^{\frac{4-x}{2}} \int_0^{\frac{12-3x-6y}{4}} dz\,dy\,dx$$

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Then rewrite the integral in the order $\int \int \int dy\,dx\,dz$ with the appropriate limits of integration.

<li style="page-break-before: always">
Find the volume of the tetrahedron bounded by the coordinate plane 
$z=4-4x-2y$.
<br><img src="0g/12.7.tet3.jpg"><br>

<li style="margin-top: 26ex;">
Find the volume of the tetrahedron shown below.
<br><img src="0g/12.7.tet4.jpg"><br>


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<li class="justme answer">First figure, by geometry $V=\frac12*(2\times1\times1)=1$; 
<br>Setting up the integral...
$$\int_{x=0}^2\int_{y=0}^1\int_{z=0}^{1-y} dz\,dy\,dx=\int_{x=0}^2\int_{y=0}^1(1-y) \,dy\,dx=...$$

<li class="justme answer">By geometry $V=4\times2\times2+4\times2\times\frac124=32$ 
<li class="justme answer">$$\begineq V&=&\iiint dV=\iiint dx\,dy\,dz\\
&=& \int_{x=0}^1\int_0^{y=2-2x}\int_0^{z=4-4x-2y}dz\,dy\,dx\\
&=&\frac43
\endeq
$$
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