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Contents

Section 6.1 - Integrating a function that blows up

Section 6.2 - Limits of integration at infinity

Section 6.1 - Integrating a function that blows up

Section 6.2 - Limits of integration at infinity

When we integrate a function that blows up to infinity at some point in the interval we're integrating, the result may be either finite or infinite.

◊ The function blows up to infinity at one end of the region of integration, but let's just try evaluating it, and see what happens.

The result turns out to be finite. Intuitively, the reason for this
is that the spike at *x*=0 is very skinny, and gets skinny fast
as we go higher and higher up.

◊ Integrate the function *y*=1/*x*^{2} from *x*=0 to *x*=1.

◊

Division by zero is undefined, so the result is undefined.

Another way of putting it, using the hyperreal number system, is that if we were to integrate from ε to 1, where ε was an infinitesimal number, then the result would be -1+1/ε, which is infinite. The smaller we make ε, the bigger the infinite result we get out.

Intuitively, the reason that this integral comes out infinite is
that the spike at *x*=0 is fat, and doesn't get skinny fast enough.

These two examples were examples of improper integrals.

Another type of improper integral is one in which one of the limits of integration is infinite. The notation

means the limit of , where *H* is
made to grow bigger and bigger. Alternatively, we can
think of it as an integral in which the top end of the
interval of integration is an infinite hyperreal number.
A similar interpretation applies when the lower limit is
-∞, or when both limits are infinite.

◊ Evaluate

◊

As *H* gets bigger and bigger, the result gets closer and closer
to 1, so the result of the improper integral is 1.

Note that this is the same graph as in example 75, but with the *x* and *y* axes
interchanged; this shows that the two different types of improper integrals really aren't so different.

◊ Newton's law of gravity states that the gravitational force between two objects
is given by *F*=*Gm*_{1}*m*_{2}/*r*^{2}, where *G* is a constant, *m*_{1} and *m*_{2} are the objects' masses,
and *r* is the center-to-center distance between them. Compute the work that must be done
to take an object from the earth's surface, at *r*=*a*, and remove it to *r*=∞.

◊

The answer is inversely proportional to *a*. In other words, if we were able to start from
higher up, less work would have to be done.

**1**.
Integrate

or show that it diverges.

**2**.
Integrate

or show that it diverges.

**3**.
Integrate

or show that it diverges.

**5**.
Integrate

or show that it diverges. (Problem 15 on p. 99 suggests a trick for doing the indefinite integral.)

**6**.
Prove that

converges, but don't evaluate it.

**7**.
(a) Verify that the probability distribution *dP*/*dx* given in example 60 on page 80
is properly normalized.

(b) Find the average value of *x*, or show that it diverges.

(c) Find the standard deviation of *x*, or show that it diverges.

**8**.
Prove

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.