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Home Applications of the Integral Improper Integrals Other Types and Summary  
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Other Types and Summary
There are other types of improper integrals. If f is continuous on the halfopen 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 halfopen 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. DEFINITION A function f is said to be piecewise continuous on an interval I if f is defined and continuous at all but perhaps finitely many points of I. In particular, every continuous function is piecewise continuous. 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
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.


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