Chapter 6. Improper integrals
6.1 Integrating a function that blows up
When we integrate a function that blows up to infinity at some point in the interval we're integrating, the result may be either finite or infinite.
Example 1◊ Integrate the function y=1/√x from x=0 to x=1.
◊ The function blows up to infinity at one end of the region of integration, but let's just try evaluating it, and see what happens.
The result turns out to be finite. Intuitively, the reason for this is that the spike at x=0 is very skinny, and gets skinny fast as we go higher and higher up.
◊ Integrate the function y=1/x2 from x=0 to x=1.
Division by zero is undefined, so the result is undefined.
Another way of putting it, using the hyperreal number system, is that if we were to integrate from ε to 1, where ε was an infinitesimal number, then the result would be -1+1/ε, which is infinite. The smaller we make ε, the bigger the infinite result we get out.
Intuitively, the reason that this integral comes out infinite is that the spike at x=0 is fat, and doesn't get skinny fast enough.
These two examples were examples of improper integrals.