what i do all day
somebody asked me recently what i do for my project, and as usually happens, about halfway through trying to explain what it was about, I realised what it actually is about, and how to explain it.
i am trying to prove (actually have proved, now) is a statement about integration, which is basically measurement of shapes. the shapes i am trying to measure are two-dimensional ones, so the whole project is about trying to estimate or measure plane area.
now, the area of a rectangle is defined as width x height. this something which is taken as a given rather than proved, and forms the basis of all other ideas of area. for instance, we know the area of a triangle is 1/2 base x height because for any triangle we can draw a rectangle with these dimensions which the triangle takes up exactly half of.
now suppose we're measuring a shape like a circle which doesn't have straight sides. there's no real way of breaking it down into rectangles or pieces of them. so a good procedure goes like this: draw a grid over the shape. add up the area of all the boxes which are completely contained within the shape. this provides an approximation of the area. if you need a better approximation, do it again with a smaller grid. of course, to the extent that it's approximate, it will be an underestimate, because the area we're measuring is completely contained within the shape.
shapes which like circles, have smooth boundaries, are 'nice' in that we can get as accurate as we like, by choosing smaller and smaller grids. one way to make sure that we are measuring something called 'area' accurately and not just pullling results out of thin air is to apply the opposite procedure: draw a grid over the shape and add up the area of each box which contains any part of the shape. again, this estimate should get better and better with smaller grids. but in this case, we'll overestimate the area, and then our better estimates will decrease towards the area.
what we're hoping when we do this is that the 2 estimates will get closer and closer together, and the true area will be between the two. watch this space to find out why this isn't always as easy as it sounds, and what i'm doing about it..
I believe I am missing the point. Why are you using rectangles to find the area of a circle when you should be using pie r squared?
Please explain or my head may explode in the process of looking for deeper meanings.
-pi r squared is fine if you know about it (and know what pi is). but before pi was invented, you needed a process like this to define what the area of a circle actually is, before you can work out what pi is.
-i probably shouldn't have used a circle for the example, but this method works for any shape with smooth edges; circles, ovals, general blobs.
i like your name by the way.
Thanks for explaining... luckily the head exploding incident I mentioned in my previous post has now been avoided. It's an interesting activity you created.
i'm glad you feel the same way about it.
You ever do stochastic calculus? No more smooth boundaries. Ugh.
It mostly comes up in Mathematical Finanace (my grad background), but probably shows up here and there in other topics (I can't immediately think of one).
The Wikipedia page on Stochastic Calculus mentions all the big buzzwords and concepts, but doesn't really explain them.
Depending on who teaches the subject and how it's taught, Stochastic Calculus can be reasonbly straightforward and very useful, or can be laughably difficult and of no use whatsoever. If the students don't have a really solid background in measure theory (true in most Math Finance classes, as there is usually all manner of business school detritus mixed in with math students), then it's definitely a waste of time to be real rigorous. Frankly, (in this topic, but not necessarily other math topics) I think it's sort of a waste to be super rigorous anyway. It's one of the few topics where a deeper understanding isn't very useful.