Besides the p-test, there are a few basic principles that go into the rest of this section.
If we have a convergent integral and we make its interval of integration smaller, the new integral will also converge.
Oddly enough, we can also make the interval of integration larger, as long as we don't include any points where the function is badly behaved. If
converges and f isn't badly behaved on [a, b], then
also converges.
This is because we changed the original improper integral by sticking on a finite area, and a definite integral
that isn't improper is always finite.
On the other hand, if
diverges then
will diverge also.
The moral of the story is that when we're looking at integrals of the form
,
it doesn't really matter what the lower limit of integration is so long as f (x) is well behaved on the interval of integration.
The integral
will converge if and only if
converges, if and only if
converges.
Sample Problem
We know that
converges. This means
also converges, because we've only added on a finite amount of area.
Sample Problem
We know that
converges. This means
also converges. The integral
is finite, so we've only increased the area by a finite amount.
If
0 ≤ f (x) < g(x)
for all x in [a, b] at which both functions are defined, and
converges, then
must converge as well. The area under g on [a, b] includes the area under f on [a, b], so if the area under g is finite the area under f must be finite as well. Similarly, if 0 ≤ f (x) < g(x) for all x in [a, b] at which both functions are defined, and
diverges, then
must diverge as well. The area under g on [a, b] includes the area under f on [a, b], so if the area under f is infinite the area under g must be infinite as well.
If 0 ≤ f (x) < g(x) for all x in [a, b] at which both functions are defined, and
converges, that doesn't help us figure out what the corresponding integral of g does. It doesn't help us to know that the area under g is larger than a finite area. Similarly, knowing
diverges doesn't help us figure out what the corresponding integral of f does. It doesn't help us to know that the area under f is smaller than an infinite area.
This property also extends to improper integrals with infinite limits.
If 0 ≤ f (x) < g(x) for all x in [a, ∞), then
If the integral of g is finite, so is the integral of f. If the integral of f is infinite, so is the integral of g.
Example 1
Given the graph below, does
converge or diverge?
|
and the x-axis on [1,∞) has finite area. Call this region B.
Example 2
Given the graph below, does
converge or diverge?
|
diverges, which means the region under 
on the interval (0,1] has infinite area. Call this region A.
Example 3
Given the graph below, does
converge or diverge?
|
diverges, which means the area under 
on the interval (0, 1] is infinite.
Exercise 1
Look at the picture below, which shows a region A within a region B.
If the area of region A is infinite, what does that tell us about the area of region B?
Exercise 2
Look at the picture below, which shows a region A within a region B.
If the area of region B is finite, what does that tell us about the area of region A?
Exercise 3
Look at the picture below, which shows a region A within a region B.
If the area of region A is finite, what does that tell us about the area of region B?
Exercise 4
Look at the picture below, which shows a region A within a region B.
If the area of region B is infinite, what does that tell us about the area of region A?
Exercise 5
From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
, but since
and
and
doesn't help, because
Exercise 6
From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
is the same as the function 
. We know the integral of this function from 0 to 1 converges, and the integral of this function from 1 to ∞ diverges.
and
that doesn't help us at all.
, we can't tell what
and 
Exercise 7
From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
on (0,1] and 
on [1,∞) and
on [0,1), but the integral
on [0,1), but 
on [1,∞).
on [1,∞).
Exercise 8
If 0 < f (x) and
converges, does
converge or diverge?
Exercise 9
Use the graph to determine if 
converges or diverges.
, and
Exercise 10
Does 
converge or diverge?
