Type I Improper Integral:
exists for every number 
, then
provided the limit exists as a number.
exists for every number 
, then
provided the limit exists as a number. In either case above, the improper integrals are called convergent if the corresponding limits exist, and divergent otherwise.
and 
are
convergent, then we define
where the choice of a is completely arbitrary.
Type II Improper Integral:
provided the limit exists as a number.
provided the limit exists as a number. In either case above, the improper integrals are called convergent if the corresponding limits exist, and divergent otherwise.
and 
are convergent, then we define
is convergent if p>1 and divergent if 
.
Comparison theorem: Suppose that f and g are continuous functions
with 
for 
.
is convergent, then so is
.
is divergent, then so is
.
It is easy to be tricked into trying to integrate over a discontinuity: that's the one that's easiest to miss. Watch out!
In section 8.8 we encounter another strange fact about infinity: though a
region may be unbounded, its area may be finite. This can happen in (at least)
two different ways: either the domain may head off to 
, or the
function may have a vertical asymptote. Both of these cases give rise to what
are called ``improper'' integrals. How delicate sounding!
Both cases are made concrete, are defined, through the use of limits. Similar to the use of partial sums in section 12.2, we see that ``partial areas'' are used, and if the limits exist, we say that the improper integrals exist.
We can use comparison, much as we did in section 12.1 and 12.2: if one
unbounded region is contained within another, and the larger is the area
corresponding to a convergent improper integral, then the first region's area
is finite (its improper integral converges).