Calculus Small Changes, Big Ideas

Section 7.1 Area between curves

• Area bounded by \(y=f(x)\le g(x)\) over \(a\le x\le b\text{:}\)

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

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• Area bounded by \(x=f(y)\le g(y)\) over \(c\le y\le d\text{:}\)

  \begin{equation*} A=\int_{c}^{d}(g(y)-f(y))\dd{y} \end{equation*}

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• Polar area bounded by one curve:

  \begin{equation*} A=\frac{1}{2}\int_{\alpha}^{\beta} r^{2}\dd{\theta} \end{equation*}

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• Polar area bounded by two curves:

  \begin{equation*} A=\frac{1}{2}\int_{\alpha}^{\beta} (r\tsub{outer}^{2}-r\tsub{inner}^{2})\dd{\theta} \end{equation*}

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• If the demand curve \(p=D(q)\) and the supply curve \(p=S(q)\) meet at the equilibrium point \((\bar{q},\bar{p})\text{,}\) then the consumer surplus is

  \begin{equation*} CS=\int_{0}^{\bar{q}}(D(q)-\bar p)\dd{q}\text{,} \end{equation*}

  and the producer surplus is

  \begin{equation*} PS=\int_{0}^{\bar{q}}(\bar p-S(q))\dd{q}\text{.} \end{equation*}

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• A Lorenz curve is a function \(L(x)\text{,}\) where \(0\le x\le 1\text{,}\) that describes how wealth is distributed among the members of a population. Given any Lorenz curve \(L\text{,}\) the Gini coefficient is defined as

  \begin{equation*} =\frac{1}{2}\int_{0}^{1}(x-L(x))\dd{x}\text{.} \end{equation*}

  This can be used as a quantitative measure of income inequality.

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