roni said:
Why "we don't actually need division"?
For a
Riemann-Integral (usually intended when we refer to
integral) we need to be able to evaluate the sum of rectangular areas:
$$\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right)$$
and take it to the limit.
As you can see this requires subtraction, multiplication, and addition, but not division.
That means that if we would have the axioms of a
Ring instead of a
Field, that the integral would still be well-defined.
(A
Field has all axioms that a
Ring has and more.)
Btw, I've just realized that we actually need an
Ordered Field (or
Ordered Ring), since we also need the comparison operator $<$ to define a partition $\{x_i\}$ with $x_i < x_{i+1}$, and we need it for the definition of a
limit as well.
Either way, to define the
derivative, we need to be able to evaluate:
$$\frac{f(x)-f(a)}{x-a}$$
Therefore we need
division.
Note that at this point I'm leaving out the more exotic definitions of integrals (such as the
Lesbesgue-Integral) and derivatives (such as the
Formal Derivative).
