**Motivation:** Consider a function *f* defined on a closed interval
*a*,*b*].*f* is
continuous and positive. Then it makes sense to look at the region between
the *x*-axis and the graph of *f*.

If we can somehow determine the area of this region, we will call this number
the **definite integral** of *f* from *a* to *b*.

There are many ways to try to determine the area. Depending on the properties
of the function *f*, it may be difficult or even impossible to do so.
Here we will try the approach of Riemann. It is based on a simple observation
that the area of a rectangle is easy to calculate. Therefore we try to
approximate the region under the graph of *f* by suitable rectangles.

It seems from the picture that if we made the rectangles really narrow, the
error of approximation would be small. By taking narrower and narrower
rectangles, with a little bit of luck the resulting approximated areas
converge to some number, namely the area of the region under the graph of
*f*. This procedure can break down if the approximation errors do not
get smaller for narrower rectangles; this depends on the shape of *f*,
for really wild functions the region is strange and it may not make any sense
to talk about its area.

Now we will make this procedure precise.

The widths of rectangles are determined by splitting the interval
*a*,*b*]*partition*:

Definition.

Consider a closed interval[ By aa,b].partitionof[ we mean any finite seta,b]of points from P={x_{0},x_{1},...,x_{N}}[ such thata,b]a=x_{0}<x_{1}< . . . <x_{N}=b.

Assume that we have a bounded function *f* on an interval
*a*,*b*].*a*,*b*]*N* segments that determine the sides of the approximating
rectangles:

Now we have to decide on their heights. There are several methods, here we
use the one that is easiest to handle. To be on the safe side, we look at the
largest and smallest possible (and reasonable) rectangles, obtaining the
*upper sum* and the *lower sum*:

Definition.

Letfbe a bounded function defined on a closed interval[ Given a partitiona,b].Pof[ fora,b],set k= 1,...,N,We define the

upper sumassociated withPbyWe also define the

lower sumassociated withPby

Note that since the function *f* is bounded, the suprema and infima in
the definition always exist finite. Therefore the sums make sense. In both
sums we are adding areas of rectangles. Their bases are given by the
partition, the heights by the supremum or infimum of *f* in each
rectangle.

The upper and lower sum is shown in the following pictures. On the left, the area of the shaded region is the upper sum; on the right, the area of the shaded region is the lower sum. We also indicated expressions connected with calculating the area of the third rectangle.

If we denote the area under the graph of *f* by *A* (hoping that it
makes sense), then from the picture it seems clear that

To determine the area *A* we will try to manipulate the rectangles so
that the upper sum gets smaller and the lower sum gets larger, until they get
almost equal. Since the area is always between the upper and lower sum,
equality between the two sums means that we determined *A*. This
manipulation has the form of taking narrower rectangles. The error of
approximation is then smaller, which means that the upper sum gets smaller
(and therefore closer to *A*) and the lower sum gets larger (and closer
to *A*). In the next picture, compare the error of approximation of the
upper and lower sum when we refine the partition.

The advantage of the upper/lower sum approach is that we do not have to worry
about mechanics of this procedure, all the details are hidden in the
definition below. Unfortunately, the Riemann approach using rectangles
succeeds only if the function *f* is nice enough, when *f* is
*Riemann integrable*. Precisely:

Definition

Letfbe a bounded function defined on a closed interval[ We say thata,b].fisRiemann integrableon[ if the infimum of upper sums through all partitions ofa,b][ is equal to the supremum of all lower sums through all partitions ofa,b][ a,b].Then we define the

Riemann definite integralofffromatobby

We usually just say **Riemann integral**, it is understood that we mean
the definite integral. Since for Riemann integrable functions, the infimum of
upper sums is equal to the supremum of lower sums, we could also use the
latter to determine the Riemann integral.

The snaky shape is called the **integration sign**, it is in fact a very
elongated S (for *sum*). The integrated function is sometimes called the
**integrand**. We have the **lower limit** *a* and the **upper
limit** *b*, giving the integrating interval
*a*,*b*].*x* is called **dummy variable** because it is not really important.
Since the Riemann integral is related to the area under the graph of
*f*, the only important information is the shape of the graph. So if we
decide to use a different variable in the same formula, the shape and
therefore the integral stay the same. Thus, for instance,

Indeed, the area under the same piece of the given parabola is always the same, regardless of what letter we write next to the horizontal axis.

The symbol

For a more thorough explanation of the meaning of the integral notation (not mathematically correct, but very useful for understanding the concept), click here.

In our definition, we put the smaller limit (the left endpoint) as a lower
limit. Sometimes we may want to "integrate backward", from *b* to
*a*. Often we want to be able to simply write the integral without
worrying about the order, so we need a more general definition. This
is done as follows: Let *a* < *b*.

Now that we can integrate with any order of limits, the above equation becomes a general rule: We can switch the limits in the integral, provided we also add the minus sign in front.

**Example:** We investigate the integral

We need to decide on some partitions that would involve smaller and smaller
segments, hoping that the corresponding upper and lower sums will get closer
until they agree. Unless there is a good reason to do otherwise, it is
usually a good idea to try a regular partition, that is, given a natural
number *N*, split the interval *N* equal
segments. Thus we have the following partition (check)

Now we need to determine the suprema and infima for the sums, but this should be easy just by looking at a picture:

We can calculate the upper and lower sum:

Since our partitions are very specific, one can not expect that in general
they would already give us the area (in the definition, we investigated
all possible partitions). However, here the function is very nice and it
turns out that when we send *N* to infinity, the sums approximate the
area well. We have to be a little bit more careful to do it properly by
definition:

Thus

that is,

Hence the function
*f* (*x*) = *x* + 1

We can check that this answer is correct by direct calculation from the picture using the formula for the area of a trapezoid.

The above calculation was not easy, even though we were lucky that we
remembered the formula for adding first *N* natural numbers. For more
complicated functions, it may be impossible to determine an explicit formula
for the upper and lower sums. This is the reason why we usually use other
means than the definition for evaluating Riemann integrals (see
The Fundamental Theorem of
Calculus).

In our pictures we always had a positive function; the Riemann integral is
then equal to the geometric area of the region between the graph of *f*
and the *x*-axis. What if we have a negative function? Since the value
of *f* determines the height of rectangles, we get areas with negative
sign. Therefore, for negative functions, the Riemann integral is equal to
minus the geometric area of the region between the graph of *f* and the
*x*-axis.

For a general function, the Riemann integral is equal to the **mathematical
area** of the region between the graph of *f* and the *x*-axis,
that is, the geometric area of the parts above the *x*-axis minus the
geometric area of the parts below the *x*-axis. For instance, in the
following picture, *A* with subscripts 1,2,3 denote geometric areas
of the individual parts:

This is a very important question. For our purposes, the most useful fact is this:

Theorem.

Every continuous function on a closed interval is Riemann integrable on this interval.

This example shows that if a function has a point of jump discontinuity, it may still be Riemann integrable. On the other hand, the example of Dirichlet function shows that if there is too many points of discontinuity, the function is not Riemann integrable. In fact, a function defined on a closed interval is Riemann integrable there exactly if it does not have too many points of discontinuity. For more information, click here.

The fact that Riemann integrability is not hurt by a finite number of
discontinuities is related to the fact that the value of Riemann integral is
not influenced by a change of the integrated function at a finite number of
points. Precisely, assume that *f* is Riemann integrable on an interval
*a*,*b*].*g* is a function that is equal to
*f* on *a*,*b*]*g* is also Riemann integrable on
*a*,*b*]*f* and *g*
agree.

To see why this could be true, look at the following picture, where we
obtained *g* by adding 1 to the function *f* at the point *c*.

The region under the graph of *g* is the same as the region under the
graph of *f*, plus an extra vertical segment at the point where we
changed *f* into *g*. Since this segment has the thickness of one
point, which is zero, its area is also zero and therefore there is no extra
area under *g*.

We will close this section with one useful statement:

Theorem.

Every monotone function on a closed interval is Riemann integrable on this interval.

Properties of the Riemann integral

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