Subsections

• Purpose
• Getting Started
• Background
  • Unbounded integrands
    • Examples
  • Unbounded intervals of integration
    • Examples
  
  
• Probability Density Function
• Exercises

Improper Integrals

Purpose

The purpose of this lab is to use Maple to introduce you to the notion of improper integral and to give you practice with this concept by using it to prove convergence or divergence of integrals involving unbounded integrands or unbounded intervals or both.

Getting Started

To assist you, there is a worksheet associated with this lab that you can copy into your home directory by going to your computer's start menu and choose run. In the run field type:

\\filer\calclab

when you hit enter, you can then choose MA1023 and then choose the worksheet

Improper_int_start_B11.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.

Background

Our basic theorem for $\int_{a}^{b} f(x)\,dx$ is that the integral exists if $f(x)$ is continuous on the closed interval $[a,b]$. We have actually gone beyond this theorem a few times, and integrated functions that were bounded and had a finite number of jump discontinuities on $[a,b]$. However, we don't have any theory to help us deal with integrals $\int_{a}^{b} f(x)\,dx$ involving one or more of the following.

1. Functions $f(x)$, for example rational functions, that have vertical asymptotes in $[a,b]$ (or are not bounded on $[a,b]$).
2. Integrals where the interval $[a,b]$ is unbounded, for example intervals like $[a, \infty)$, $(-\infty, b]$, or $(-\infty, \infty)$.

We have already seen at least one example of the problems you can run into if the function is unbounded. Recall the clearly absurd result

$$\int_{-1}^{1} \frac{1}{x^2} \, dx = -2$$

that is obtained by blindly applying the FTOC. The second type of problem, where the interval of integration is unbounded, occurs often in applications of calculus, such as the Laplace and Fourier transforms used to solve differential equations. It also occurs in testing certain kinds of infinite series for convergence or divergence, as we will learn later.

We start with the following definition.

Definition 1   We say that the integral

$$\int_{a}^{b} f(x) \, dx$$

is improper if one or both of the following conditions is satisfied.

1. The interval of integration is unbounded.
2. The function $f(x)$ has an infinite discontinuity at some point $c$ in $[a,b]$. That is, $$\lim_{x \rightarrow c} f(x) = \pm \infty$$.

Unbounded integrands

To see how to handle the problem of an unbounded integrand, we start with the following special cases.

Definition 2   Suppose that $f(x)$ is continuous on $[a,b)$, but $$\lim_{x \rightarrow b^{-}} f(x) = \pm \infty$$. Then we define

$$\int_{a}^{b} f(x) \, dx = \lim_{t \rightarrow b^{-}} \int_{a}^{t} f(x) \, dx ,$$

provided that the limit on the right-hand side exists and is finite, in which case we say the integral converges and is equal to the value of the limit. If the limit is infinite or doesn't exist, we say the integral diverges or fails to exist and we cannot compute it.

Definition 3   Suppose that $f(x)$ is continuous on $(a,b]$, but $$\lim_{x \rightarrow a^{+}} f(x) = \pm \infty$$. Then we define

$$\int_{a}^{b} f(x) \, dx = \lim_{t \rightarrow a^{+}} \int_{t}^{b} f(x) \, dx ,$$

provided that the limit on the right-hand side exists and is finite, in which case we say the integral converges and is equal to the value of the limit. If the limit is infinite or doesn't exist, we say the integral diverges or fails to exist and we cannot compute it.

Cases where $f(x)$ has an infinite discontinuity only at an interior point $c, a <c < b$ are handled by writing

$$\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$$

and using the definitions to see if the integrals on the right-hand side exist. If both exist then the integral on the left-hand side exists. If either of the integrals on the right-hand side diverges, then $\int_{a}^{b} f(x)\,dx$ does not exist.

Examples

Here is a simple example using Maple to show that $$\int_{0}^{2} \frac{1}{x} \, dx$$ doesn't exist.

> ex1 := int(1/x,x=a..2);

> limit(ex1,a=0,right);

The example above used the right option to limit because the right-hand limit was needed. If you need a left-hand limit, use the left option in the limit command. Maple can usually do the limit within the int command.

> int(1/x,x=0..2);

Unbounded intervals of integration

These are handled in a similar fashion by using limits. The definition we need the most is given below.

Definition 4   Suppose $f(x)$ is continuous on the unbounded interval $[a, \infty)$. Then we define

$$\int_{a}^{\infty} f(x) \, dx = \lim_{t \rightarrow \infty} \int_{a}^{t} f(x) \, dx ,$$

provided the limit on the right-hand side exists and is finite, in which case we say the integral converges and and is equal to the value of the limit. If the limit is infinite or fails to exist we say the integral diverges or fails to exist.

The other two cases are handled similarly. You are asked to provide suitable definitions for them in one of the exercises.

Examples

Using the definition for $$\int_{2}^{\infty} \frac{1}{x^2}$$.

> ex2:=int(1/x^2,x=2..a);
> limit(ex2,a=infinity);

This command shows that Maple takes the limit definition into account in the int command.

> int(1/x^2,x=2..infinity);

Probability Density Function

Definition 5   We say that a random variable $X$ is continuous if there is a function $f(x)$, called the probability density function, such that

i

$f(x) \geq 0$, for all $x$

ii

$\int_{- \infty}^{\infty} f(x) \, dx = 1$

iii

$P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$ where $P(a \leq X \leq b)$ represents the probability that the random variable $X$ is greater than or equal to $a$ but less than or equal to $b$.

Exercises

1. The gamma function is an example of an improper integral often used to approximate non-integer factorials and is defined below:
   
   $$\Gamma (t) = \int_0^\infty x^{t-1} e^{-x} \,dx$$
   
   

   A)

   Verify that $\Gamma(n)=(n-1)!$ for $6!$.

   B)

   Verify that $$\Gamma^2(\frac{1}{2})=\pi$$

   C)

   Verify Euler's reflection formula:
   

   $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$

   
   

   You may need to use the $\texttt{simplify}$ command.

2. Another example of an improper integral is calculating the probability of a process that can be modeled as a normal distribution with a continuous random variable $x$ having mean $\mu$ and standard deviation $\sigma$. For example, the normal distribution is a probability density function widely referred to as the bell-shaped curve. The probability density function for a normal distribution with mean $\mu$ and standard deviation $\sigma$ is given by the following equation.
   
   $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{(x- \mu)^2}{2 \sigma^2}} , \mbox{ for $- \infty < x < \infty$\ }$$
   
   
   Show that $\int_{- \infty}^{\infty} f(x) \, dx = 1$. Intelligence Quotient (IQ) scores are distributed normally with mean 100 and standard deviation 15.

   A)

   What is the probability that an IQ score is above 120?

   B)

   What percentage of the population has an IQ score between 85 and 115?

   C)

   Show that the median IQ score is equivalent to the mean in this example. (Hint: Find $m$ such that $P(-\infty \leq x \leq m)=0.5$ or $P(m \leq x \leq \infty)=0.5$.)

   D)

   What would be the minimum IQ score for a person to be in the top 1 % of the population?

   E)

   What would be the maximum IQ score for a person to be in the bottom 1 % of the population?

3. Determine which of the regions described below have finite area.

   A)

   The region below $$f(x)=\frac{1}{2x^2+x-1}$$, above the $x$ axis, over the interval $[1,\infty)$.

   B)

   The region below $$f(x)=\frac{1}{\sqrt{2x^2+x-1}}$$, above the $x$ axis, over the interval $[1,\infty)$.

   C)

   The region below $$f(x)=\frac{1}{(2x^2+x-1)^{\frac{2}{3}}}$$, above the $x$ axis, over the interval $[1,\infty)$.