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Calculus Small Changes, Big Ideas

Bill Shillito

Section 24.3 Riemann and Darboux integrals

Introduction goes here.
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  • A partition of \([a,b]\) is a finite set
    \begin{equation*} P=\set{x_0,x_1,x_2,\cdots,x_n} \end{equation*}
    such that \(a=x_{0}\text{,}\) \(b=x_{n}\text{,}\) and \(x_{0}<x_{1}<x_{2}<\cdots<x_{n}\text{.}\)
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  • Given a partition \(\set{\seq x0n}\) of \([a,b]\text{,}\) for each subinterval \([x_{i-1},x_{i}]\text{,}\) we denote:
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    \begin{align*} m_i \amp = \inf\set{f(x):x\in[x_{i-1},x_i]}\\ M_i \amp = \sup\set{f(x):x\in[x_{i-1},x_i]} \end{align*}
    Now we can define the upper sum as
    \begin{equation*} U(f,P)=\sum_{i=1}^{n} M_{i}\cdot(x_{i}-x_{i-1}) \end{equation*}
    and the lower sum as
    \begin{equation*} L(f,P)=\sum_{i=1}^{n} m_{i}\cdot(x_{i}-x_{i-1})\text{.} \end{equation*}
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  • A partition \(Q\) of \([a,b]\) is a refinement of \(P\) if \(P\subseteq Q\text{.}\)
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    If so, then
    \begin{equation*} U(f,P)\ge U(f,Q)\quad\text{and}\quad L(f,P)\le L(f,Q)\text{.} \end{equation*}
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  • If \(P\) is a partition of \([a,b]\text{,}\) then \(L(f,P)\le U(f,P)\text{.}\)
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  • If \(P_{1}\) and \(P_{2}\) are any partitions of \([a,b]\text{,}\) then
    \begin{equation*} L(f,P_{1})\le U(f,P_{2})\text{.} \end{equation*}
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  • Let \(f:[a,b]\to\bb R\) be a bounded function and let \(\cl P\) be the collection of all partition of \([a,b]\text{.}\) The upper integral of \(f\) is defined to be
    \begin{equation*} U(f)=\inf_{P\in\cl P}U(f,P) \end{equation*}
    and the lower integral is
    \begin{equation*} L(f)=\sup_{P\in\cl P}L(f,P) \end{equation*}
    If \(L(f)=U(f)\text{,}\) we say that \(f\) is Darboux integrable. This is equivalent to being Riemann integrable.
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  • If \(m=\inf\limits_{a\le x\le b}f(x)\) and \(M=\sup\limits_{a\le x\le b}f(x)\text{,}\) then
    \begin{equation*} m(b-a)\le L(f)\le U(f)\le M(b-a)\text{.} \end{equation*}
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