Other Types and Summary

There are other types of improper integrals. If f is continuous on the half-open interval [a, b) then we define

If f is continuous on (-∞, b] we define

We have introduced four types of improper integrals corresponding to the four types of half-open intervals

[a, b),       [a, ∞), (a,b], (-∞, b].

By piecing together improper integrals of these four types we can assign an improper integral to most functions which arise in calculus.

We can introduce the improper integral whenever f is piecewise continuous on I and a, b are either the endpoints of I or the appropriate infinity symbol. A few examples will show how this can be done.

Let f be continuous at every point of the closed interval [a, b] except at one point c where a < c < b. We define

Example 7

Example 8

Example 9

If H and K are positive infinite hyperreal numbers and c is finite, then

H + K is positive infinite, H + c is positive infinite,

-H - K is negative infinite,

-H + c is negative infinite,

H - K can be either finite, positive infinite, or negative infinite.

By analogy, we use the following rules for sums of two infinite limits or of a finite and an infinite limit. These rules tell us when such a sum can be considered to be positive or negative infinite. We use the infinity symbols as a convenient shorthand, keeping in mind that they are not even hyperreal numbers.

∞ + ∞ = ∞,

∞ + C = ∞,

-∞ - ∞ = -∞,

-∞ + C = -∞,

∞ - ∞ is undefined.

Example 10