Definition of definite integrals

Definite Integrals

Definition (Definite integral) The definite integral from \(a\) to \(b\) of the function \(\mathrm{f} \) is defined as :

\[\displaystyle \displaystyle\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = \big[ \mathrm{F}(x) \big]_{a}^{b} = \mathrm{F}(b) - \mathrm{F}(a)\]

where \(F(x) = \displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x\) and \(a\) and \(b\) are the (lower and upper) limits of the integration.

Note that:

1. In a definite integral, there is an absence of the constant of integration.
2. In the IB syllabus, for the definite integral to be defined, we assume that the integrand \(\mathrm{f}\) is (piecewise) continuous on \the closed bounded interval \([a,b]\).
   For example,
   \[ \displaystyle \displaystyle\int_{-1}^1 \frac{1}{x} \, \mathrm{d}x \]
   is undefined.
3. The Graphic Display Calculator (GDC) is very useful in evaluate definite integrals; this is especially the case when it is difficult, or even impossible, to obtain an antiderivative of the integrand. You are warned that the radian mode be turned on whenever the integrand involves trigonometric functions.