What is the history and significance of the Riemann integral?

[Nano-Passion]

I understand the integral through the technique of using the Riemann Sum. The whole concept of the integral makes sense but what is the history and motivation behind it? Whats the big point of it. I realize that it transitions into being able to take the area of any object, given its slope of course. But I feel that there is a hole in my understanding of its importance because I'm sure there is a much richer beauty and application to it.

 

[DivisionByZro]

Read a bit from this page:

http://en.wikipedia.org/wiki/Area_of_circle#Using_polygons

The idea of approximating areas by using smaller shapes for which we know the size is an old technique, but archimedes was one of the first to use a method of exhaustion for the area of a disk.

 

[Nano-Passion]

Read a bit from this page:

http://en.wikipedia.org/wiki/Area_of_circle#Using_polygons

The idea of approximating areas by using smaller shapes for which we know the size is an old technique, but archimedes was one of the first to use a method of exhaustion for the area of a disk.

But I was asking about other applications of integration other than finding the area of a shape. Looking at integration from the point of view of finding the area is the easiest thing, but I know it has a much richer use; I just need it explained.

A bit of background, I am taking Calculus II at the moment.

 

[chiro]

Hey Nano-Passion.

One of the most important ideas of the integral is that differentiation and integration processes are linked in a fundamental way.

Basically integration allows us to sum up "infinitesimal changes" to get a final sum (I'm using the Riemann integral for this example). The power of a Riemann integral when an analytic solution exists is that we have an exact representation in terms of a F(b) - F(a) term which represents an infinite sum of "infinitesimal" changes.

Now if we can describe how something changes in the context of various measures, then we can find a way to calculate how the changes end up describing the final result of all the changes.

So think of the different kinds of things that we can model as changes: we have things like flux calculations for electricity and magnetism calculations: we have line integrals that help us figure out force calculations for all kinds of mechanical problems in physics and engineering.

Basically we have got a way to relate deltas with corresponding sums and what this means is that if we know what the deltas are, then we can come up with a model to figure out the sum.

Now this idea is old and many side ideas have come into existence like trying to find a function with specific properties (for example it might be a function that given certain conditions gives us a solution that 'minimizes' the integral). So in this way we've gone from a way to 'sum up lots of changes' to 'engineer a function with constraints that gives us these changes'.

Also thinking also in the context of 'infinitesimal changes', you have to realize that if we have a way to describing what is changing with respect to what, then we can apply calculus to analyze it. This is exactly what is done with differential geometry.

As a simple example think of the arc-length between two points on a manifold. If we were using flat space (basically the right angle geometry we are used to) then this wouldn't be a problem because we have things like the pythogorean theorem as well as things like the scalar product to do this.

But what if we are dealing with a geometry like a saddle or a circle? Then we have to turn to calculus to describe the rates of change because unlike the flat geometry, the changes are not constant: they differ depending on what point of the surface you are on and because the path is not a straight line, the problem gets more complex. In fact you might find that you can take multiple paths from one point to another that are still minimal and yet have the same distance!

If you understand it in this context, it will make sense of seeing things like differential geometry and the calculus of variations as things that build on the fundamental of the calculus.

 

[Fredrik]

I think one of the best ways to understand Riemann integrals is to think about how you would want to define "work" in Newtonian mechanics. If the force is constant, it's natural to define "work" as force·distance. This definition ensures that if you push it twice as far, or had to push it twice as hard, you have done twice the "work". So this product seems to be a good measure of how hard it's going to be to move something. But how can you generalize this definition to situations where the force varies along the path? The answer is to chop up the path in small segments, on which the force is approximately constant. Now you calculate force·distance for each segment, and add up the results. It certainly seems like this will be a good mathematical representation of the concept we already understand intuitively if the segments are short, and an even better representation if the segments are shorter. So it's very natural to generalize the definition by assigning the term "work" to the limit of a sequence of results of such calculations as the length of all segments go to zero, if that limit exists and doesn't depend on the exact details of how the limit is taken.

There are of course many other examples. One of them is that at constant velocity, the distance traveled, s, is calculated as s=vt, where v is the velocity and t is the time. But what if v varies with time? You should however only need one example, like the work example above, to see the point. The concept of "work" can't even be defined properly without integrals.