Types of Integrals

of the interval into sub-intervals , where . The width of each interval is given by

and the norm of the partition is given by . In each interval, choose a value so that using any method.

Then, we define the definite integral

This general definition is rarely useful for evaluating integrals. Instead, we usually make the following simplifications:

1. We choose the width of each interval to be the same, so that we can define for all . Sub-dividing into equal sub-intervals,

this means that

so that

as .

2. We choose the Right-Hand Riemann Sum which means we choose the

in each interval to be the right-hand endpoint of each interval. Thus for all

.

This enables us to write the Riemann sum as

STEP 1999, Math III, #3: Justify, by means of a sketch, the formula

Show that

Evaluate

The Fundamental Theorem of Calculus

Let be any antiderivative of the function . Then the Fundamental Theorem of Calculus says that

It is not obvious (at least not to me) how the antiderivative of the integrand function is produced as a result of calculating a Riemann sum. Justifying this transformation requires a very important theorem called the Mean Value Theorem. A careful derivation of the Fundamental Theorem of Calculus can be found in any good calculus textbook.

There are three basic types of integrals. All of them are related to one another but they serve different purposes.

Indefinite Integrals

An indefinite integral looks like this

This represents a FAMILY of antiderivatives of the integrand .

Examples:

1.

each of which is an antiderivative of the function .

2.

each of which is an antiderivative of the function .

Definite Integrals with constant limits

A definite integral with constant limits represents a NUMBER. The number may correspond to area, average value, volume, arclength, etc.

Examples:

1.

represents the number 1.

2.

represents the number .

Notice that with a definite integral, the end result is a number so that the variable inside the integral can be replaced with any other variable. This is why it is often referred to as a dummy variable.

Definite Integrals with one variable limit

A definite integral with one variable upper limit represents a Function. This function is precisely one of the antiderivatives of the integrand.

Examples:

1. If

is the specific antiderivative of that passes through the point .

To see this we refer to the fundamental theorem of calculus.

Furthermore

In this case, we could evaluate the integrand and show that

if we had the patience and really needed it, but it may not be necessary.

2. If

represents the specific antiderivative of that passes through the point . In this case, we could not evaluate the integrand in the usual sense since it is not possible using any techniques to find a closed form antiderivative for this function. This is really not so bad. We can find any point on

that we want using numerical techniques.

Mathematica gives the following values

, let be the area of the largest rectangle with sides parallel to the coordinate axes that can fit in the region bounded by the curve , the y-axis and the line

. Show that can be written in the form

is defined, for , by

Show that .

Making use of a sketch show that, for

,

and deduce that .

(iii) In the case

use the above to establish the inequality

for

.

Green Question: Let

Find and simplify .

STEP 2000, Math II, #5: It is required to approximate a given function , over the interval , by the linear function , where is chosen to minimise

Show that

The residual error,

, of this approximation process is such that

Show that

is small, and .]

[Try this out on the functions and ]

Average Value

When you take the average of

numbers you find their sum and divide by :

How might we find the average value of a function .

We could take a sample of

values of the function, add them up, and divide by :

However, this is only a sample. To really get the idea of the average value of a function we should continue increasing the

number of sample points and make sure our points are equally spaced within an given interval

. In fact, we should take as many as possible.

With this in mind, we would want the average value of a function to be

This should remind you of something. Looking above at the Riemann sum we can see that the average value of a function can be defined as

The Mean Value Theorem for Integrals

such that such that

, the largest value of is .

(ii) Find constants

, , and such that, for all ,

(iii) Hence, or otherwise, prove that