Huh? I would argue that differentiation is easier. You can differentiate any function, albeit perhaps with great difficulty; not all functions are integrable. (The preceding is taken from last year’s Calc BC class, so it may not be completely accurate.)
You’ve got it completely backwards. Of the functions you look at in calculus classes given by formulae, that’s correct, but in the real world it’s exactly the opposite.
Any continuous function on an interval can be integrated. In fact, it doesn’t even have to be continuous – there can be a finite number of discontinuities and you can still handle it by breaking up the interval. If you define integration the right way (hat tip: Lebesgue) you can even integrate functions with an infinite number of discontinuities like the characteristic function of the rational numbers (1 on every rational numbers, 0 on every irrational numbers).
On the other hand if you put all the continuous functions on a dart board and threw a dart, you’d hit a differentiable function with probability zero. In fact, you’d hit a function that has any point where the differential exists with probability zero. Almost all continuous functions cannot be differentiated anywhere.
To make matters worse for Archimedes, he knew what an area was, but not what a parametrized curve was. You could talk about integration by slicing up areas into smaller and smaller pieces, but you can’t even give the heuristic definition of a derivative (slope of tangent line) without analytic geometry.
Getting a workable notation for something is extremely important in mathematics, and that’s what’s most likely to vary between cultures. Arithmetic is horribly difficult using Roman numerals and is much easier using a place-value system like the Hindu-Arabic numerals, for example.
Thank you. I somehow feel better now. I always felt bad hearing about these brilliant people whose intellects could “make short work of most anything they encountered” (Funny thing is, you never got to meet these people, I now suspect kind of like you never got to meet the person who actually microwaved their dog or had a kidney stolen…).
ISTR reading somewhere that, prior to the Renaissance or so, something we take for granted in math, like that you can have an equation like y = x[sup]3[/sup] + 5x[sup]2[/sup], wouldn’t have made sense, because a squared quantity was an area and a cubed quantity was a volume- how can you make any sense of adding an area to a volume?
That’s what I was getting at. 
I think we’re getting back to the “standing on the shoulders of titans” proposition. Today’s math grad student finds calculus a breeze only because of the breakthrough understandings made before him. Another factor alluded to: Archimedes couldn’t then speak our modern mathematical lingua franca. He lacked both the notational tools and the understanding of algebra. He was trying to crack the calculus nut using geometry.
I didn’t see your (far better) response before I responded. In a sense, Archimedes lacked the Rosetta stone to decipher the mystery text before him. He wouldn’t have the keys to unlock the code. He might intuitively understand some of the concepts, after many hours of study, but couldn’t rigorously prove them or use them in a meaningful way.
That’s the biggest stumbling block in identifying the geometric constructions as abstract algebraic operations. In fact, since the Greeks only thought up to volumes even Heron’s formula was an enormous leap forward. The classical proof (in Heath’s edition of Elements, in the commentary after IV.4) requires making two areas, then considering the proportion of their squares, though a “square” really was the area of a square on a line segment. To consider the “square” on an area was a huge leap of faith in formalisms.
Actually, even Newton and Leibniz couldn’t rigorously prove them. They just used the intuitive notion of “differentials”, which is where we get the abusive notation “dy/dx” from. Limits were a far later addition to the theory in order to put analysis onto a rigorous foundation.
(to add to what Mathochist already said) You’re right if by “differentiate” and “integrate” you mean “find a formula for the derivative of” and “find a formula for the antiderivative of.” But (1) that isn’t what mathematicians mean by the word “integrable,” which, if I may put it very sloppily, has to do with whether the “area under the curve” exists*, and (2) the ideas of functions and algebraic formulas and such would have been foreign to Archiimedes.
*(If you want to get technical, whether or not a function is integrable depends on what kind of integration you’re talking about (i.e. whose definition you’re using)—Riemann, Lebesgue, etc.)