relation between integral and summation What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here  where I got two answers (one is based on summation notation) and the other is based on integral notation and I do not know yet which one to accept . So I would like to understand the connect between the two ?

The Riemann or Lebesgue integral is in a sense an continuous sum. The symbol $\int$ is adapted from a letter looking like a somewhat elongated 's' from the word summa. In the definitions of the Riemann and the Lebesgue integrals the ordinary finite summation $\sum _{k=1}^n$ is used but the relation between the two is deeper than a mere 'one is used in the definition of the other' kind of relation. 
The modern approach to viewing integrals is measure theory. Without getting into too many technical details one defines a measure space as an abstraction to the notion of measuring length or volume. Then real valued functions on a measurable space may be integrated to give a single real number. 
For a particular measure space one gets the Lebesgue integral which itself, in a suitable way, subsumes the Riemann integral. On the other hand, given any set $X$ with $n$ elements there is a measure space structure on it such that for any function $f:X\to \mathbb {R}$ the integral of $f$ with respect to that measure is precisely the sum of the values that $f$ attains. In that sense general measure theory subsumes finite sums (of real numbers). More interestingly, given any countable set $X$ there is a measure on it such that the integral of real values functions on $X$ correspond to the infinite sum of its values. (Both of these remarks are trivial.) Thus measure theory subsumes infinite sums. From that point of view, a summation corresponds to integrals on a discrete measure space and the Lebesgue or Riemann integral corresponds to integrals on a continuous measure space.
