Calculus Small Changes, Big Ideas

Section 15.4 Triple integrals

• Triple integral of \(f(x,y,z)\) over box \(B=[a,b]\times[c,d]\times[e,f]\text{:}\)

  \begin{equation*} \iiint_{B} f(x,y,z)\dd{V}=\limit{\Delta x,\Delta y,\Delta z\to 0}\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{k=1}^{p} f(x,y,z)\,\Delta x\,\Delta y\,\Delta z \end{equation*}

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• Iterated integrals can take any of six orders of \(\dd{x}\text{,}\) \(\dd{y}\text{,}\) \(\dd{z}\text{,}\) for example:

  \begin{equation*} \int_{x=a}^{x=b}\int_{y=g(x)}^{y=h(x)}\int_{z=p(x,y)}^{z=q(x,y)}f(x,y,z)\dd{z}\dd{y}\dd{x} \end{equation*}

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• Would like to find a good way to go over setting them up

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• Volume:

  \begin{equation*} V=\iiint_{D}\dd{V} \end{equation*}

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