Improper+Integrals

=__an introduction__=

What do you do when you must take the integral of a function in which somewhere inside, or at the limits of integration, the function doesn't have a finite number or seems to be discontinuous? Let's see some examples...

Taking the integral from certain values in each of these examples will result in a integral form that attempts to evaluate an area underneath the curve which goes to infinity. In the first example, taking indicates an attempt to evaluate each successive addition of area until infinity. This may seem impossible, but if the distance between the graph of f(x) and the x-axis continues to decrease, then the area may approach but never reach a certain value.

If we try to take, then the integral will attempt to evaluate the area at x=0, which goes up to seemingly infinity. Once again, this may make the integral infinity, but if the distance between the graph and the y-axis decreases sufficiently, the area may actually be approaching but never reaching a certain limit.

For both of these examples, the improper integral must be used. **In short, improper integrals are used when an integral appears to be adding on an infinite amount of area under the curve. The improper integral will determine whether or not that area truly is infinite as area is added, or if the area reaches a limit.**

=__definition__= __An improper integral is an integral in which one of the limits is not finite, or because at some point c between the limits of integration, f(c) is discontinuous.__

To evaluate improper integrals, we must use limits in this form:

This integral properly evaluates example (a) from the introduction. When the integral is improper because there exists a point c where f(c) is not defined, such as in example (b), we must break up the integral into two expressions, each one evaluating the integral from one of the limits to the point c.

This integral properly evaluates example (b).

=__results of a properly evaluated improper integral__= = Indeed, one possible result is that the improper integral diverges. That occurs when, given the integral, the area simply continues to accumulate to infinity (or negative infinity) within the limits of integration. Otherwise, the improper integral may converge onto a certain value instead of forever adding more area. The textbook uses a good comparison between two functions that appear to approach zero as x goes to infinity:. . If we take the integral of both these functions as x approaches infinity, it seems as if both will simply continue to add a smaller amount of area, towards an infinite value. But as we'll see, the area underneath the curve of f(x) will never reach a certain value; the addition of "slices" under the curve will decrease in size so quickly that a certain limit will never be reached. f(x) can be said to converge onto a certain value (which we will find shortly). g(x) diverges because the area under the curve continues to increase to a larger number no matter how far "into infinity" we take x. = =__evaluating improper integrals__=

Improper integrals are evaluated just like a normal integral. Find the antiderivative of the function to be integrated, and then evaluate the antiderivative at the limits of integration, subtracting the values.

Example 1:

After evaluating this integral, we can conclude that the integral converges onto 800. No matter how far into infinity the limit of integration is taken, the area that is added will never exceed 800.

Example 2:





We know that as x goes to infinity in the function ln(x), the function will continue to go to infinity. Therefore, after evaluating this integral, we can conclude that the integral diverges, since it does not approach a finite value.

=__handy video resources__= = = This YouTube poster uploads numerous videos on improper integrals... media type="youtube" key="85-HNJyuyrU" height="385" width="480" media type="youtube" key="KvC4XyayEC0" height="385" width="480"media type="youtube" key="Q_VSj0sDA5I" height="385" width="480"

Cited Works... Just the textbook and my brain!

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