One knows generally that integrals have to do with summations, but when do we need to use integrals as opposed to finite sums or infinite summations (infinite series)? Sometimes there is a vague idea that integrals are used to sum functions that are continuous, while summations are for discrete values. We will iron out these points in this article.
A function maps points in a domain to a range. Here, we consider real-valued functions – that is, functions whose range is a subset of 
Now, the domain could have finite number of elements, countably infinite number of elements or uncountably infinite number of elements. (Countably infinite set is one where the elements can be listed in a sequence one after another. Enumeration of the set is possible. Examples of countably infinite sets are 
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When the domain for a function has finite number or countably infinite number of points it is possible to sum the values of the function across all points in the domain. It consists merely of listing out the values of the function for each point of the domain and adding them. In case of countably infinite number of points, the sum turns out to be an infinite series. Of course, the infinite series may converge or diverge, but the sum can still be written out and it is meaningful to ask the questions of convergence/divergence and what the sum converges to if it converges.
In case, the domain has uncountably infinite number of points, it is impossible to sum the values of the function for all points of domain as it is impossible to list the values of the function one after another. This is where integral’s role is. In the concept of integral, the domain is divided into subsets. While the total number of points in a domain is uncountably infinite, the number of subsets is often manageable. Now, instead of summing the values of the function at each point of the domain, we sum the average values of the function for all the subsets.
We are not discussing all the details of integration. One needs to study advanced calculus for Riemann integration and real analysis for Lebesgue integration. For example, how do we make the subsets? Assume the domain is a subset of real line. In Riemann integral, nearby points in the domain get to one subset as long as the values of the function at points in one subset do not differ too much. In Lebesgue integral, points in the domain for which the values of function are nearby get into one subset. Then, how is the average calculated over the subsets? In Lebesgue integration, what is called a measure is defined on the domain. And, the value of the function over the subset (remember the function values do not differ much in each subset) is multiplied by the measure of the subset to give its average. In Riemann integration, it is the same except that length of the intervals is used as a measure and since points in each subset of the domain are nearby (in one interval), this measure works well here.
The idea of dividing the domain of a function into subsets is easily comprehensible for functions such as step functions (these are constant over intervals). It is not so for functions whose values “continually vary”. The theory of integration addresses how integration can be carried out for functions that vary continually. There are, of course, functions that cannot be integrated, but integration is possible for a wide variety of functions. In particular, Lebesgue integration is more sophisticated than Riemann integration, in that it enables a much broader class of functions to be integrated.
In summary, integrals enable us to carry out the sum of values of a function defined on a domain with uncountably many number of points. Theory of integration addresses how a variety of functions including many of those that vary continually can be integrated.