Short and Sweet Calculus

6.2 Properties of the definite integral

In this section, we discuss some key properties of definite integrals that will help us evaluate definite integrals in an easy way.

  1. \({\displaystyle \int_{a}^{b}[f(x)+g(x)]dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx}.\)

  2. \({\displaystyle \int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)}dx,\) where \(c\) is a constant.
    In particular \({\displaystyle \int_{a}^{b}-f(x)dx=-\int_{a}^{b}f(x)dx}.\)

  3. \({\displaystyle \int_{a}^{b}cdx=c(b-a)}\), where \(c\) is a constant.

  4. \({\displaystyle \int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx},\) either \(c\) is between \(a\) and \(b\) or outside.

The truth of Properties (1)–(3) immediately follows from the definition of the definite integral and the summation properties, namely \[[f(x_{1}^{*})+g(x_{1}^{*})]\Delta x+\cdots+[f(x_{n}^{*})+g(x_{n}^{*})]\Delta x=\left[f(x_{1}^{*})\Delta x+\cdots+f(x_{n}^{*})\Delta x\right]+\left[g(x_{1}^{*})\Delta x+\cdots+g(x_{n}^{*})\Delta x\right]\] and \[cf(x_{1}^{*})\Delta x+\cdots+cf(x_{n}^{*})\Delta x=c\left[f(x_{1}^{*})\Delta x+\cdots+f(x_{n}^{*})\Delta x\right].\] The truth of Property (3) can be shown graphically (see Figure 6.4).

Example 6.3. Find \(\int_{8}^{9}f(x)dx\), if we know that \(\int_{-1}^{9}f(x)dx=-2\) and \(\int_{-1}^{8}f(x)dx=4\).

Solution

Using property 4, we have \[\underbrace{\int_{-1}^{9}f(x)dx}_{-2}=\underbrace{\int_{-1}^{8}f(x)dx}_{4}+\int_{8}^{9}f(x)dx.\] Therefore, \[\int_{8}^{9}f(x)dx=-6.\]

  1. If \(f(x)\geq0\) for \(a<x<b\) then \[\int_{a}^{b}f(x)dx\geq0.\]

  2. If \(f(x)\geq g(x)\) for \(a<x<b\) then \[\int_{a}^{b}f(x)dx\geq\int_{a}^{b}g(x)dx.\]

Property (5) says if the graph of \(f\) does not lie below the \(x\)-axis, the signed area of the region lying between the curve \(y=f(x)\) and the \(x\)-axis is nonnegative. The truth of property (6), because \(f(x)-g(x)\geq0\), follows from Properties (5), (1), and (2).


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