Subsections

• Purpose
• Getting Started
• Improper Integrals
  • Unbounded integrands
    • Examples
  • Unbounded intervals of integration
    • Examples
  
  
• Sequences and Series
• 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. You will also be introduced to to the concepts of convergence and divergence of sequences and series.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples similar to some of the exercises.On your Maple screen go to File - Open then type the following in the white rectangle:

\\storage\academics\math\calclab\MA1023\Improper_int_start_A14.mw

You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read through the exercises, start up Maple, load the worksheet, and go through it carefully. Then you can start working on the exercises.

Improper Integrals

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);

Sequences and Series

An infinite sequence is a list of numbers defined by the formula $a_n$ where $n$ is the set of positive integers ${1,2,3,...}$. A sequence is said to converge if the numbers in the in the list approach a single value as $n$ approaches $\infty$. If the numbers in the list increase or decrease without bound as $n$ gets larger, then we say that the sequence diverges.

An infinite series is the sum of an infinite sequence. More precisely, the sum of an infinite series is defined as the limit of the sequence of the partial sums of the terms of the series, provided this limit exists. If no finite limit exists, then we say that the series is divergent.

The sum of an infinite series is defined as $$\lim_{n \rightarrow \infty} S_n$$, where $S_n$ is the partial sum of the first n terms of the series. However, because of the algebraic difficulty of expressing $S_n$ as a function of n, it is not always possible to find sums by directly using the definition.

Exercises

1. The gamma function is an example of an improper integral often used to approximate non-integer factorials and is defined below:
   
   $$\Gamma (t) = \int_0^\infty x^{t-1} e^{-x} dx$$
   
   
   Evaluate $\Gamma(t)$ by calculating the improper integral for $t=\{1,2,5/2,3,7/2,4,5\}$ and for each integer value of $t$, check your answer by calculating $(t-1)!$.

2. Both of the following improper integrals given below do not exist. Show, by calculating a limit, why they do not exist.

   A

   
   

   $$\int_0^{\infty} sin(x) dx$$

   
   

   B

   
   

   $$\int_{-2}^{2} \frac{1}{(x^2 - 4)} dx$$

   
   

3. Plot $\frac{1}{x}$ on the interval $1 \leq x \leq 100$. Recall from Calculus II that the volume of a solid of revolution formed by rotating $f(x)$ about the $x$-axis over the interval $[a,b]$ is
   
   $$\pi \int_a^b f(x)^2 dx .$$
   
   
   and the surface area is
   
   $$2\pi \int_a^b f(x)\sqrt{1+f'(x)^2} \, dx .$$
   
   

   Find the volume and surface area of the solid obtained by revolving the curve $$y = \frac{1}{x}$$ about the $x$-axis, between $x = 1$ and $x = \infty$.

4. 1. Write the sequence $$\frac{7n^2-2}{4n^2+9}$$ as a function. List the first fifteen terms.
   2. Plot the first fifteen terms. Does it appear to converge or diverge? If it seems to converge, to what does it converge?
   3. Calculate the limit of the sequence as $n$ approaches $\infty$. Is your answer close to your approximation above?
   4. Write a function for the sum of the sequence. List the first ten terms of the sequence of partial sums.
   5. Plot the at least the first ten terms of the sequence of partial sums (You may want to restrict the $y$ range to $0 \leq y \leq 10$). Does it appear that the series converges or diverges? If it seems to converge, to what does it converge?
   6. Calculate the limit of the series as $n$ approaches $\infty$.