Riemann Integration
Before continuing with methods of antidifferentiation, let’s consider another geometric problem: integration. Here’s an example:
We’ve got a function whose graph is drawn in red, and we want to find the area contained between the graph, the -axis, and the two blue lines at
and
. We’ll approximate this by cutting up this interval into
pieces and choosing a sample point
in each piece, like so:
Now we’ve just got a bunch of rectangles, and we can add up their areas to get
where is the value of the function at the
th sample point, and
is the width of the
th strip. Now as we cut the strips thinner and thinner, our stairstep-like approximation to the function should get closer and closer to the real function, and our approximation to the area we’re interested in should get better and better.
So how can we formalize this process? First, let’s take an interval and think about how to cut it up the strips. We do this by picking a collection of points
. We get a bunch of smaller intervals
, and in each one we pick some
. This structure we call a “tagged partition” of the interval
. We define the “mesh” of a partition to be its thickest subinterval,
, and we’ll want to somehow take this down to zero.
We can now see that the collection of all the tagged partitions of an interval form a directed set! We say that a tagged partition is a “refinement” of a tagged partition
if every partition point
is one of the
, and every tag
is one of the
. That is, we get from
to
by splitting up some of the slices of
and adding new tags to the new slices. Then we define
if
is a refinement of
. This makes the collection of tagged partitions into a partially-ordered set.
To show that this is a directed set, consider any two tagged partitions and
, and make a new partition by using all the partition points from each one. Now look at each slice in the new partition. It can’t have more than one
tag or
tag, so it has either zero, one, or two distinct tags. If it has no tags, add one. If it has one tag, do nothing. If it has two distinct tags, split it between them (notice how we’re using the topology of
to say we can make this split). At the end, we’ve got a new partition that refines both of
and
. And thus we have a directed set.
Now if we have a function on
, we can get a net on this directed set. Given any tagged partition
, we define the “Riemann sum”
Finally, we say that the function is “Riemann integrable” if this net converges to a limit
, and in this case we define the “Riemann integral” of
:
which is, at last, the area under the curve as we set out to find.