*improper integrals*, and how the limits to find them make sense.

The point of this page is to look at *improper integrals*,
and how the limits to find them make sense.

Let's consider the function *f*(*x*) =
1/(*x*^{2} - 1).
If we want to find either of the integrals

then, we have an improper integral: in the first case the
*y*-range is unbounded, and in the second the *x*-range is
unbounded.

Let's consider the second of these: we want to find the area shown in the figure to the right. However, because one of the dimensions of the graph is unbounded, we can't just apply our usual fundamental theorem of calculus methods: it doesn't make to say

for some antiderivative *F*(*x*) of *f*(*x*),
because infinity isn't a point and we can't evaluate a function
there.

So instead we consider a well defined finite domain by stopping the
integral at some point *b*, as shown in figure 2 to the
right. This integral we can do (using partial fractions or a table):
we have

Then, we know this is the same as the original integral in the limit
as *b* goes to infinity. Thus

Note that it is not true that the integral with the upper limit of
infinity is the same as the integral with the upper limit of *b*:
the two are equal only as *b* goes to infinity:

So, if you're writing out the solution, either stop and work the
integral with *b* in the limit (making a note in your solution
that you're considering a different integral), and then say later that
the integral you want is the limit of your result---or write limits
the whole way through:

To find

Find:

slicing volumes

Last modified: Tue Jan 29 15:55:29 EST 2008

Comments to:glarose(at)umich(dot)edu

©2008 Gavin LaRose, UM Math Dept.