Integral of Discontinuous Functions

In computing definite integrals one must first make sure that the function to be integrated is continuous on the interval. For instance,

Incorrect:

This is clearly wrong because 1/x² > 0 so the area under the curve cannot be negative. The mistake is that 1/x² is undefined at x = 0 and hence the function is discontinuous at x = 0. Therefore the area under the curve and the definite integral

are undefined (Figure 4.3.6).

Figure 4.3.6