What is integration?
Roughly speaking, integration is the inverse of differentiation, but there are several ways to think about it...
Given a suitably well behaved function
At any particular point
Integration covers a lot more cases than just Real valued functions of Real numbers. You can integrate over any kind of measurable set - e.g. a plane, a curve, a surface, a volume. The function that you are integrating may have any kind of value that is possible to sum and multiply by a scalar, e.g. Real, Complex, vector.
In such contexts you can think of an integral as a sort of infinite sum of values of a function over infinitesimally small pieces of the set over which you are integrating.
For example, suppose you have a function
#(int_(p in S) f(p) dp) / A#
If we split the surface of the sphere into a large number of little patches
#int_(p in S) f(p) dp ~~ sum_i A_i f(p_i)#