Calculus Small Changes, Big Ideas

Section 15.1 Double integrals and iterated integrals

• Define the double integral of \(f(x,y)\) over a rectangular region \(D=[a,b]\times[c,d]\text{:}\)

  \begin{equation*} \iint_{D} f(x,y)\dd{A}=\limit{\Delta x,\Delta y\to 0}\sum_{i=1}^{m}\sum_{j=1}^{n} f(x,y)\,\Delta x\,\Delta y \end{equation*}

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• Include some simple by-hand approximations with the rectangular region broken into small squares or rectangles

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• Fubini’s theorem says that we can write such a double integral as an iterated integral under certain ‘niceness’ conditions (will have to find later):

  \begin{equation*} \iint_{D} f(x,y)\dd{A}=\int_{c}^{d}\int_{a}^{b} f(x,y)\dd{x}\dd{y}=\int_{a}^{b}\int_{c}^{d} f(x,y)\dd{y}\dd{x} \end{equation*}

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• Work out some iterated integrals

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• If \(f(x,y)=g(x)\cdot h(y)\text{,}\) then

  \begin{equation*} \iint_{D} f(x,y)=\int_{a}^{b} g(x)\dd{x}\cdot \int_{c}^{d} h(y)\dd{y} \end{equation*}

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• Area:

  \begin{equation*} A=\iint_{D} \dd{A} \end{equation*}

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• Average value:

  \begin{equation*} \frac{\DS\iint_{D} f(x,y)\dd{A}}{(b-a)(d-c)}=\frac{\DS\iint_{D} f(x,y)\dd{A}}{\DS\iint_{D} \dd{A}} \end{equation*}

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