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?