What is integration?

Given a suitably well behaved function #f(x):RR->RR#, and an interval #(a, b)#, the definite integral #int_a^b f(x) dx# is the "area under the curve" between #a# and #b#.

At any particular point #t in RR#, the rate of change of area as you increase #t# is equal to the #f(t)#. That is, the derivative of the integral is equal to the original function.

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 #f(p)# defined for points on the surface #S# of a sphere, with surface area #A#. Then the average value of #f(p)# over the surface of the sphere is:

#(int_(p in S) f(p) dp) / A#

If we split the surface of the sphere into a large number of little patches #S_i# of areas #A_i#, each containing a representative point #p_i in S_i#, then we could approximate the integral over the surface:

#int_(p in S) f(p) dp ~~ sum_i A_i f(p_i)#