Definite integrals

Summary

• Throughout the section, \(f\) is a continuous function defined on \(I=[a,b]\) with \(a<b\) .
• If \(f(x)≥0\) for all \(x∈I\) then we define

\[\int_{a}^{b}{ ‍ }f(x)dx\]

• as the area of the shape bounded by the graph of the function, the \(x\) -axis and the vertical lines through \(x=a\) and \(x=b\) .

• Notation:
• The symbol \(\int{ ‍ }\) is called the integral sign.
• The function inside the integral between the integral sign and the symbol \(dx\) is called the integrand .
• \(a,b\) are called the (lower and upper) limits of the integral.
• \(dx\) indicates that \(x\) is the variable of integration.
• \(\int_{a}^{b}{ ‍ }f(x)dx\) is always a real number.
• \(x\) is a dummy-variable and \(\int_{a}^{b}{ ‍ }f(x)dx\) = \(\int_{a}^{b}{ ‍ }f(u)du\) .
• If \(f(x)≤0\) for all \(x∈I\) then we define

\[\int_{a}^{b}{ ‍ }f(x)dx\]

• as minus the area of the shape bounded by the graph, the \(x\) -axis, \(x=a\) and \(x=b\) .
• In general, when \(f\left( x \right)\) takes both positive and negative values, we define \(\int_{a}^{b}{ ‍ }f(x)dx\) as the area above the \(x\) -axis minus the area below the \(x\) -axis.
• For convenience, we define

\[\int_{b}^{a}{ ‍ }f(x)dx=-\int_{a}^{b}{ ‍ }f(x)dx\]

• and for any \(c∈(a,b)\) we define

\[\int_{c}^{c}{ ‍ }f(x)dx=0\]

• This is not how \(\int_{a}^{b}{ ‍ }f(x)dx\) is defined formally . Actually, there are several different definitions of \(\int_{a}^{b}{ ‍ }f(x)dx\) . However, with the given assumptions, all definitions agree and they also agree with our informal understanding of area. Also, continuity is a sufficient but not necessary for the integral to exist.