Subsections

• Introduction
• Background

Definite Integrals

Introduction

In Exercise 1 we look at the definite integral as the limit of Riemann sums (see Sections 5.3 to 5.5 of the text). In Exercise 2 we show how Maple can, in many cases, give us antiderivatives (see Section 5.1). In Exercise 3 we use Maple to implement the amazing theorem about definite integrals given in Section 5.6 - do you know what theorem this is? Exercise 4 gives you an antiderivative problem with two different looking answers. In Exercise 5 we take a look at the kind of approach that could be used on a problem for which there was no practical way to apply that amazing theorem from Section 5.6.

Background

For Exercise 1 ....

It is not claimed that the following procedure is efficient. Rather, it is presented as an educational demonstration in the hope that you will develop a greater appreciation of how the definite integral arises as a limit of Riemann sums.

We use a right sum with n subintervals to set up a Riemann sum. Then we simplify this sum and use summation techniques to reduce the Riemann sum to a function of n. Finally, the limit as n goes to infinity is taken.

Here are Maple commands to carry out the steps outlined above for the given g on the interval [2,5]. Notice the use of the sum command. More information about sum can be found in Maple's online help.

  > with(student):

  > g:=x->x^2+2*x-4;

  > rightsum(g(x),x=2..5,n);

  > expand(``);

  > R:=(12/n)*sum(1,i=1..n)+(54/n^2)*sum(i,i=1..n)+(27/n^3)*sum(i^2,i=1..n);

  > expand(R);

  > limit(``,n=infinity);

For Exercise 2 ....

In Maple the int command can be used to obtain antiderivatives. All you need to do is list the function and the variable of integration. Here is a sample of the syntax

  > int((sin(x))^2,x);

For Exercise 3 ....

The int command can also be used to evaluate definite integrals. If possible, Maple will find the needed antiderivative and then apply the Fundamental Theorem of Calculus. Here is a sample of the syntax.

  > int((sin(x))^2,x=0..Pi);

For Exercise 4 ....

A theorem mentioned on the first day of class will be useful here.

Exercise 5 ....

When a usable antiderivative cannot be found, a definite integral then has to be approximated by one of what are called numerical integration techniques. Here we will approximate a definite integral by using a midpoint rule Riemann sum (see Lab 1).

An approximation to a quantity is usually not of much use unless we have some idea of by how much the approximation might be ``off'' from the actual value of the quantity. We now discuss such an error bound for midpoint rule approximations.

The midpoint rule with n subintervals (designated as M_n) usually gives better accuracy than either the left endpoint rule (L_n) or the right endpoint rule (R_n). This means that, for a given n, M_n is generally closer to A, the value of the definite integral under discusion, than either L_n or R_n. In numerical analysis texts it is shown that the error, EM_n, in using M_n to approximate the area under y = f(x) on [a,b] satisfies

where B is the global maximum of $\mid f^{\prime\prime}(x)\mid$on [a,b]. In practice, B is often approximated by a number K that is an upper bound for B, that is B < K. For instance, if $f(x) = \sin 2x$ on $\left[0,\frac{\pi}{5}\right], K$ might be taken as 4. Do you see why? For more complicated functions, Maple can be used to get a value for K that is close to B. Note that the error bound formula gives a worst case estimate, the accuracy achieved for a given number of subintervals n may be much better than the guarantee given by the formula.

In Section 10.3 (part of MA 1023) you can see two additional numerical integration techniques. Note that when, for a definite integral, you use evalf with int, Maple invokes a powerful numerical approximation routine.

The function g given in this exercise is not the kind of function for which a numerical integration technique is required since it has a ``nice'' antiderivative. However, it yields a good illustration of the technique under discussion and also permits us to make a useful comparison in part (d) of the exercise.