TITLE: On the exactness of the calculus formulas
QUESTION [5 upvotes]: Are calculus formulas, differentiation and  integration, exact formulas or are some approximations involved? 
That is, is the value of a definite integral of a function the exact value of the (signed) area between the graph and the axis or is it only any approximation for it, and likewise for the value of the derivative and the slope of the tangent at this point.
The question arises as when introducing these notions approximations are often  central, say, via Riemann sums. It is now not clear if the definite integral is another (sometimes more viable) way to get an approximation or if it is exact and the same for derivative and slope of the tangent.

REPLY [5 votes]: Given $a<b$ and a continuous function $f:\>[a,b]\to{\mathbb R}$ the integral
$$\int_a^b f(x)\>dx$$
is a clear cut real number, i.e., a certain element of the set ${\mathbb R}$. This number can be familiar to you, like ${7\over 13}$, $\sqrt{5}$, or $\pi$. But maybe it has never before "occurred" in mathematics.
It is the definition of this number as a limit of Riemann sums that uses approximations; but once this definition is adopted there is no question of some fishy "approximation" involved anymore.
There remains, however, the following problem: Your function $f$ can be a simple analytic expression, like $f(x):=e^{-x^2/2}$, and your scientific pocket calculator does not have the primitive of this $f$ in store, as it is not "elementary". As a consequence your calculator can only output a numerical approximation to the integral
$$\int_0^1 e^{-x^2/2}\>dx\ ,\tag{1}$$
and cannot give a value in terms of "standard" functions and constants, like $\exp$, $\cos$, $\pi$, $\sqrt{2}$, etc. Nevertheless the expression $(1)$ defines a certain real number $\xi$ exactly, i.e., to "infinitely many" decimal places.