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?
 A: If a function is continuous on some closed interval then the two integrals will agree, hence a distinction is not necessary. Otherwise I believe the context should be enough.
A: There are only two situations where it could be unclear:


*

*The integral is on a compact interval, $f$ is not Riemann integrable and is Lebesgue integrable;

*The integral is not on a compact interval, $f$ is improperly Riemann integrable and not Lebesgue integrable. 


(Note that e.g. $\int_0^1 \ln(x) dx$ is "not on a compact interval" in this sense.)
The first situation is really not a problem, because either:


*

*the discontinuities of $f$ have positive measure, in which case you're dealing with a Lebesgue integral;

*or it doesn't matter, because both exist and are the same.


The second situation requires careful attention to context. For instance, one should pay careful attention to properly understand what "$\int_1^\infty \frac{\sin(x)}{x} dx$ exists" means.
A: 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.
