What is a surface integral?
A surface integral is a generalisation of integration from linear intervals to two dimensions.
Limit of a sum
If a function is defined on a linear interval, then we can approximate the integral - the 'area under the curve' - by splitting our interval into several subintervals, evaluating the function at some point in each of the subintervals, multiplying that value by the length of the subinterval and summing. The actual integral is the limit as we make our subintervals smaller and smaller.
Analogously, if a function is defined on a surface
Integral over bounded surface
Generalisation to unbounded surface