Calculus Small Changes, Big Ideas

Section 15.2 Double integrals over general regions

• Can use a ‘bounding box’ approach to take double integrals over regions with more general regions. If \(D\) is a region over which \(f\) is defined, let \(R\) be a rectangle containing \(D\text{,}\) and let

  \begin{equation*} F(x,y)=\begin{cases}f(x,y) \amp \text{if }(x,y)\in D\\ 0 \amp \text{if }(x,y)\in R\setminus D\end{cases} \end{equation*}

  Then \(\DS\iint_{D} f(x,y)\dd{A}=\iint_{R} F(x,y)\dd{A}\text{.}\)

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• If \(D\) is bounded below by \(y=g(x)\) and above by \(y=h(x)\text{:}\)

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

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• If \(D\) is bounded on the left by \(x=g(y)\) and the right by \(x=h(y)\text{:}\)

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

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• Changing order of integration:

  \begin{equation*} \int_{x=0}^{x=1}\int_{y=x}^{y=1}\sin(y^{2})\dd{y}\dd{x}=\int_{y=0}^{y=1}\int_{x=0}^{x=y}\sin(y^{2})\dd{x}\dd{y} \end{equation*}

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