Calculus Small Changes, Big Ideas

Section 9.2 Numerical integration

• Frame this as how we can deal with functions with no closed-form antiderivative, like the ones we just saw.

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• Left Rectangular Approximation:

  \begin{equation*} L_{n}=\sum_{i=1}^{n} f(x_{i-1})\,\Delta x_{i} \end{equation*}

  If \(f\) is increasing/decreasing, this will be an under/overestimate.

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• Right Rectangular Approximation:

  \begin{equation*} R_{n}=\sum_{i=1}^{n} f(x_{i})\,\Delta x_{i} \end{equation*}

  If \(f\) is increasing/decreasing, this will be an over/underestimate.

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• Midpoint Rectangular Approximation:

  \begin{equation*} M_{n}=\sum_{i=1}^{n} f(\bar x_{i})\,\Delta x_{i} \end{equation*}

  (Here, \(\bar x_{i}=\dfrac{x_{i-1}+x_{i}}{2}\) is the midpoint of \(x_{i-1}\) and \(x_{i}\text{.}\))

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  If \(f\) is concave up/down, this will be an under/overestimate.

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  Error if \(|f''(x)|\le K\) for \(a\le x\le b\text{:}\) \(|E_{M}|\le\dfrac{K(b-a)^{3}}{24n^{2}}\)

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• Trapezoidal Approximation:

  \begin{equation*} T_{n}=\sum_{i=1}^{n} \dfrac{f(x_{i-1})+f(x_{i})}{2}\,\Delta x_{i}=\dfrac{L_{n}+R_{n}}{2} \end{equation*}

  If \(f\) is concave up/down, this will be an over/underestimate.

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  Error if \(|f''(x)|\le K\) for \(a\le x\le b\text{:}\) \(|E_{T}|\le\dfrac{K(b-a)^{3}}{12n^{2}}\)

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• Simpson’s rule:

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}\approx\dfrac{b-a}{6}\left(f(a)+4f\left(\tfrac{a+b}2\right)+f(b)\right) \end{equation*}

  Relation to trapezoid and midpoint rules: \(S_{2n}=\dfrac{T_{n}+2M_{n}}{3}\)

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  Error if \(|f^{(4)}(x)|\le K\) for \(a\le x\le b\text{:}\) \(|E_{T}|\le\dfrac{K(b-a)^{5}}{180n^{4}}\)

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