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Introduction to complex analysis

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# 1.2 Integration on the real line

We wish to define the Riemann integral of a continuous real function in such a way that if is positive on some interval , then the integral of  from to  is the area under the graph of  between and . This is illustrated by the shaded part of Figure 8. To do this, we first split the interval into a collection of subintervals called a partition.

Figure 8 Area under the graph of between and

## Definitions

A partition of the interval is a finite collection of subintervals of ,

for which

The length of the subinterval is .

We use to denote the maximum length of all the subintervals, so

Given a partition of , we can approximate the area under the graph of between and by constructing a sequence of rectangles, as shown in Figure 9.

Figure 9 Approximating the area under a graph using a sequence of rectangles

Here the th rectangle has base and height (so the top-right corner of the rectangle touches the curve). The area of the rectangle is (see Figure 10). Note that we could equally have chosen the rectangle to be of height for any point in , and the theory would still work. This is because, for a continuous function , the difference between one set of choices of values for , , and another disappears when we take limits of partitions. We have chosen merely for convenience.

Figure 10 Rectangle of height and width

Summing the areas of all the rectangles gives an approximation to the area under the graph. This sum is called the Riemann sum for , with respect to this particular partition. (You may have seen upper Riemann sum and lower Riemann sum defined slightly differently elsewhere.)

## Definition

The Riemann sum for with respect to the partition

is the sum

We now calculate the Riemann sum for a particular choice of function and partition, and then ask you to do the same for a second function.

## Example 1

Let , where . Show that for

we have

and determine .

### Solution

Each of the subintervals of has length . Therefore

Using the identity

we obtain

as required.

Finally, since is a basic null sequence, we see that

Now try the following exercise, making use of the identity

## Exercise 1

Let , where . Show that for

we have

and determine .

Each of the subintervals of has length . Therefore

as required.

Since is a basic null sequence, we see that

The Riemann sums of Example 1 approximate the area under the graph of  between and . The approximation improves as increases, and we expect the limiting value to actually be the area under the graph. However, to be sure that this limit gives us a sensible value, we should check that for any sequence of partitions of such that . The following important theorem, for which we omit the proof, provides this check.

## Theorem 1

Let be a continuous function. Then there is a real number such that

for any sequence of partitions of such that .

We can now define the Riemann integral of a continuous function.

## Definition

Let be a continuous function, where . The value  determined by Theorem 1 is called the Riemann integral of over , and it is denoted by

The theorem tells us that to calculate the Riemann integral of over , we can make any choice of partitions for which and calculate . Thus the calculation of Example 1 really does demonstrate that

We define the Riemann integral when , as follows.

## Definitions

Let be a continuous real function.

If , and is contained in the domain of , then we define

Also, for values of in the domain of , we define

As we have discussed, for a continuous real function that takes only positive values on , where , the Riemann integral

measures the area under the graph of between and . If we no longer require to be positive, then the integral still has a geometric meaning: it measures the signed area of the set between the curve , the -axis and the vertical lines and , where we count parts of the set above the -axis as having positive area, and parts of the set below the -axis as having negative area, as illustrated in Figure 11.

Figure 11 Signed area determined by the graph of between and