I'm asking this because in many applications where there is a discrete case of some calculation (that includes a summation) and another continuous case, in the continuous one we simply swap the summation sign for an integral.
Note: for the sake of disambiguation, I'm not referring to the Riemann Sum.
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Yes. In fact, if we consider Lebesgue integration, we can view sums as a specific example of an integral (namely, integration wrt counting measure over some countable set).
In probability theory, this also means that we can often prove statements for both continuous and discrete random variables at the same time, by viewing the sums we find in the discrete case as integrals as well. This makes the theory a lot more elegant compared to having to repeat the proofs and definitions for the continuous case and discrete case separately.
The short answer is - it depends. There are lots of cases where this is a healthy way of viewing the integral, and it is indeed the initial motivation behind the idea of Riemann sums. But in some cases, you need to be more careful while taking "the continuum limit".
This is especially important in problems in statistical mechanics and fluid dynamics, where you sometimes approximate discrete systems as continuous ones. For instance, there is a very standard derivation in statistical mechanics that shows the existence of a phenomenon known as Bose-Einstein condensate, in which if you take the continuum limit by turning a sum into an integral, you get an incorrect result.
But yeah, generally, in many cases it is fine and even a good way to view the idea of an integral (just wanted to also point out that it might be inaccurate at times, and should be done with caution and with proper understanding of the problem).