Improper Integrals
An improper integral is an integral of one of two types: either the interval over which the interval is taken is unbounded, or the integrand becomes unbounded in a neighborhood of one or more points in the interval (including the endpoints). The common thread in these two types of integrals is that both have the potential to be infinite.
The first type of integral is an integral over either a half-infinite interval, i.e. from a finite value a to infinity or from negative infinity to a, or over the interval from negative infinity to positive infinity. Let us first consider the first of these possibilities. Suppose f is a function that is defined for all x greater than or equal to a. We are trying to measure an area whose "horizontal dimension" is infinite. The mathematical method for doing computations involving infinity is to take a limit of finite quantities. For any number b greater than a, the integral of f from a to b (that is, the area under the graph of f between x=a and x=b) is a well-defined, finite quantity. If there is in fact a finite total area under the graph of f from a to infinity, then as b gets larger and larger the area between a and b will approach the total area from a to infinity. Therefore we define the integral of f from a to infinity to be the limit as b approaches infinity of the integral of f from a to b, if this limit exists.
To look at a concrete example, consider the integral of the function f(x) = 1/(x2) from, say, 1 to infinity. For any finite b > 1, the integral of f from 1 to b equals 1 - 1/b. As b approaches infinity, this quantity approaches 1, and hence the improper integral from 1 to infinity is defined to be equal to 1. In other words, there is one unit of area under the graph of y = 1/(x2) to the right of x=1. To understand how this can be, look at this graph. Although the graph stretches infinitely far to the right, it is shrinking rapidly as it does so. Intuitively speaking, an improper integral over an infinite interval will be finite if, on its graph, the height shrinks "faster" than its width grows. For an example of a function that shrinks as x approaches infinity but does not shrink fast enough, consider the integral of f(x) = 1/x from 1 to infinity. The integral from 1 to b equals ln(b), which approaches infinity as b approaches infinity. Thus the improper integral does not exist.
An example of an improper integral does not exist even though the area in question does not become infinite is the integral of sin x from 0 to infinity. As b approaches infinity, the area under y=sin x between 0 and b perpetually fluctuates between 0 and 1, and consequently the limit does not exist.
An improper integral from negative infinity to some finite value a is defined in the analogous way, by taking limits of proper integrals as their left endpoints approach negative infinity. An improper integral from negative infinity to positive infinity is defined by taking any value a and adding the values of the improper integrals from negative infinity to a and from a to positive infinity, provided both exist.
Now suppose f is a function, defined on a half-open interval (a,b], whose values approach infinity as x approaches a. In contrast to the integrals considered above, here we are trying to measure an area whose horizontal dimension is finite but whose vertical dimension is infinite. Intuitively, whether or not this area is finite will depend on the rate of growth of the function f as x approaches a. Formally, the integral of f from a to b is defined by computing the integral of f from c to b, where c is a value between a and b, and taking the limit as c approaches a. For example, the integral of f(x) = x(-1/3) from 0 to 1 is improper because f(x) approaches infinity as x approaches 0. For any c between 0 and 1, the integral of f from c to 1 equals (3/2)(1 - c(2/3)). As c approaches 0, this quantity approaches 3/2, so this is the value of the improper integral.
The definition is similar for an integral that is improper at its right endpoint instead of its left. For an integral from a to b that is improper at both endpoints, a value c is chosen between a and b, and the improper integrals from a to c and from c to b are added together, provided they both exist. Finally, an integral from a to b can also be improper at a value c between a and b. In this case, once again the improper integrals from a to b and from c to b are added, if they both exist.
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