Calculus Small Changes, Big Ideas

Section 3.2 Properties of integrals

• Linearity properties:

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  \begin{align*} \int_a^b [f(x)+g(x)]\dd{x} \amp = \int_a^b f(x)\dd{x}+\int_a^b g(x)\dd{x}\\ \int_a^b cf(x)\dd{x} \amp = c\int_a^b f(x)\dd{x} \end{align*}
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• Inequality property: If \(f(x)\le g(x)\) when \(a\le x\le b\text{,}\) then

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}\le\int_{a}^{b} g(x)\dd{x} \end{equation*}

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• Additive Interval Property:

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}+\int_{b}^{c} f(x)\dd{x} = \int_{a}^{c} f(x)\dd{x} \end{equation*}

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• Use the idea of functional equations to play with the Additive Interval Property to find other properties.

  • Zero Interval Property:

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    \begin{align*} \int_a^a f(x)\dd{x}+\int_a^b f(x)\dd{x} \amp = \int_a^b f(x)\dd{x}\\ \int_a^a f(x)\dd{x} \amp = 0 \end{align*}
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  • Rewind Property:

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    \begin{align*} \int_a^b f(x)\dd{x} + \int_b^a f(x)\dd{x} \amp = \int_a^a f(x)\dd{x} = 0\\ \int_b^a f(x)\dd{x} \amp = -\int_a^b f(x)\dd{x} \end{align*}
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