they are not inverse operations, differentiation, not being one to one cannot possibly be invertible. poeple often label integration as anti-differentiation, though.
as for why the fundamental theorem of calculus is true (that if f is continuous and F(x) is the integral of f from a to x where a is some constant, then F is differentible and the derivative is f) what it is saying geometrically is that if we tak f, this continuous function, and look at the rate at which the area it defines changes then that is f itself. which verbally seems quite reasonable.
but anyway, it is a formal consequence of the definition and as with most scientific results you won't bet very far if you use to many "causal" ideas.
then again i suppose we could explain it by giving a good definition of the derivative rather than just "the slope", ie one that is often times more useful and is how we ought to define it.
let f be a function, and suppse that it is differentiable, then the derivative f' is a function with the following property:
f(x+d) = f(x)+d f'(x) + junk that behaves like d^2 or worse.
ie if we change x by a small amount d then we near as damnit change f(x) by d times the derivative at x.
what is the integral of a function g from a to b? it is approximately g(a)d +g(a+d)d +g(a+2d)d+...+g(b)d
where we spit the interval fom a to b into lots of little subintervals each of width d and estimate by this sum. now surely you can see that as d gets small in both examples how it is that the derivative of the integral of g, or the integral of the derivative of f might be linked?
