Read this section to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1-5.
This important question was finally answered in the 1850s by Georg Riemann, a name that should be familiar by now. Riemann proved that a function must be badly discontinuous to not be integrable.
Every continuous function is integrable
If is continuous on the interval
then is always the same finite number, , so is integrable on .
In fact, a function can even have any finite number of breaks and still be integrable.
Every bounded, piecewise continuous function is integrable.
If is defined and bounded for all in and continuous except at a finite number of points in ,
then is always the same finite number, , so f is integrable on .
The function in Fig. 9 is always between –3 and 3 (in fact, always between –1 and 3) so it is bounded , and it is continuous except at 2 and 3. As long as the values of f(2) and f(3) are finite numbers, their actual values will not effect the value of the definite integral, and
Practice 5: Evaluate . (Fig. 10)
Fig. 11 summarizes the relationships among differentiable, continuous, and integrable functions: