When writing the integral sign $\int$, how does one know what integral is being discussed? We have the Lebesgue integral and the Riemann Integral. Generally, must the integral sign $\int$ refer to one or the other exclusively or does it depend on the integrand? Can someone provide intuition to this concept?
For example, in college-level Calculus courses, what integral is actually being used here?

It is possible that some courses may differ, but Riemann integration seems to be
fairly standard in your basic first textbook of integral calculus.
The standard curriculum for real analysis as I recall it was you get one year
of single-variable differential and integral calculus, 
then a year of multivariate calculus, and if you continue on from that
(which you will almost certainly not do unless you are majoring in math)
you get Lebesgue integration.
I think the reason may be it's relatively easy to explain Riemann integration;
to motivate Lebesgue integration you'd have to understand things like the reason
the set of rationals has measure zero and the set of irrationals does not.
The online course description for 
Calculus I at the University of Houston,
for example, mentions Riemann sums immediately under the heading of integration.
MIT's open courseware Calculus lecture notes
pretty clearly are using Riemann integration for the definite integral.
