Define "Geometry"

[Aedl Foxe]

This is a question of opinion. ... Unless, of course, it isn't, in which case it isn't. But I think it is.

My Calculus professor gave us a problem recently which is not horribly difficult to solve using methods of integration available to a second-year Calculus student. However, he challenged us to make an attempt to solve the problem geometrically.

I will be asking my professor later to clarify, but the distinction which he made intrigued me. The first method of solution which came to mind was the Method of Exhaustion utilized by Archimedes et al. which, though inarguably a geometric method, is not all that dissimilar to polar integration. In fact, the exact variation of the Method of Exhaustion I was going to use is equivalent (almost) to the Trapezoidal Rule that I learned back in my first year of Calculus.

Of course, I was also throwing around trig functions in my method, which is taught (in America, at least) in a separate class form Geometry, the year after students learn Geometry... but Archimedes and Euclid "proved theorems that are equivalent to modern trigonometric formulae" using what they would have called geometry. (Wikipedia)

My question is, then, this: Where does the difference lie between the various simple fields of mathematics? Algebra, Geometry, Trigonometry, Calculus... are these fields entirely distinct? Or is there some overlap which defies exact differentiation?

[gfauxpas]

In my personal experience, there's a lot of blurring of concepts in concepts treated in analytic geometry. For example, the graph of a function is not the same as the function itself. Nevertheless, you may have been exposed to the idea of a definite integral as the signed area of a plane region bounded by the x-axis and the graph of f. Similarly, the set of real numbers is sometimes called "the real number line", vectors might be represented by a set of directed line segments, etc.

The analytic definitions of all these things, though, don't require geometry at all. A function/matrix/coordinate/etc are well defined without drawing any pictures. So I would argue that any time you use graphs or xy-planes or whatever, you're using a geometric construct. If your teacher asks you to evaluate the unsigned area of a circle, I'd argue it's geometry. If you really wanted to be particular you would have to distinguish between the unsigned area of a circle and the limit of a riemann sum involving x^2 + y^2 = r^2 or whatever.