Chapter 6. Improper integrals

6.1 Integrating a function that blows up

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.

Example 1

◊ Integrate the function $y=1/sqrt{x}$ from x=0 to x=1.

◊ 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.

$int_0^1 x^{-1/2}:dx = left.2x^{1/2}right|_0^1$

$= 2$

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.

improper-a

a / The integral $int_0^1 dx/sqrt{x}$ is finite.

Example 2

◊ Integrate the function y=1/x² from x=0 to x=1.

◊

$int_0^1 x^{-2}:dx = left.-x^{-1}right|_0^1$

$= -1+frac{1}{0}$

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.

improper-b

b / The integral $int_0^1 dx/x^2$ is infinite.

These two examples were examples of improper integrals.

6.2 Limits of integration at infinity

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

$int_a^infty f(x):dx$

means the limit of $int_a^H f(x):dx$ , 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.

Example 3

◊ Evaluate

$int_1^infty x^{-2}:dx$

◊

$int_1^H x^{-2}:dx = left.-x^{-1}right|_1^H$

$= -frac{1}{H}+1$

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 74, but with the x and y axes interchanged; this shows that the two different types of improper integrals really aren't so different.

improper-c

c / The integral $int_1^infty dx/x^2$ is finite.

Example 4

◊ Newton's law of gravity states that the gravitational force between two objects is given by F=Gm₁m₂/r², where G is a constant, m₁ and m₂ 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=∞.

◊

$W = int_a^infty frac{Gm_1m_2}{r^2} : dr$

$= Gm_1m_2 int_a^infty r^{-2} : dr$

$= -Gm_1m_2 left.r^{-1}right|_a^infty$

$= frac{Gm_1m_2}{a}$

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.

Homework Problems

1. Integrate

$int_0^infty e^{-x} dx qquad ,$

or show that it diverges.

2. Integrate

$int_1^infty frac{dx}{x} qquad ,$

or show that it diverges.

3. Integrate

$int_0^1 frac{dx}{x} qquad ,$

or show that it diverges.

4. Integrate

$int_0^infty x^2 2^{-x} :dx qquad ,$

or show that it diverges. (solution in the pdf version of the book)

5. Integrate

$int_0^infty e^{-x}cos x : dx$

or show that it diverges. (Problem 14 on p. 97 suggests a trick for doing the indefinite integral.)

6. Prove that

$int_0^infty e^{-e^x} : dx$

converges, but don't evaluate it.

7. (a) Verify that the probability distribution dP/dx given in example 59 on page 78 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

$int_0^infty e^{-x}x^n:dx = n! qquad .$