TITLE: Is the Riemann Integral good for anything?
QUESTION [11 upvotes]: Math people:
I think it is a good idea to teach beginning calculus students the Riemann Integral (I refer to what calculus books call the "Riemann Integral" and ignore any controversy about whether it should be called the Darboux integral, etc.), since it gives a rigorous definition to the idea of "area under a curve" and you don't need measure theory to define it.  But is the Riemann integral ever used in "real" mathematics by mathematicians who know Lebesgue integration?  Is there any purpose in proving difficult-to-prove properties of the Riemann integral (say, in a real analysis class)?   The Lebesgue integral is more powerful and has those properties, plus more.
The only uses I can think of are: (i) using it as as trick to evaluate certain infinite sums or limits and (ii) a jumping-off point for learning the Riemann-Stieltjes integral.
EDIT: I just discovered a similar, unanswered question at Are Specific Facts about the Riemann Integral Logically Required? .   If you read it, you may understand my question better.

REPLY [3 votes]: The Riemann integral is good for at least two things:

Definition to be used in logical arguments; and
Numerical integration.

In courses where logical rigor is not expected, I'd omit Riemann's definition of the integral.  One could merely say it's the area under a curve.
But one approach to intuitive definition says that rather than using an integral to find an area, you use and area to approximate an integral.  An integral is a sum of infinitely many infinitely small quantities $f(x)\,dx$, and $f(x)$ could be in units of force and $dx$ in units of distance so that $f(x)\,dx$ is in units of energy, and it's not about area under a curve at all.  But draw the graph of $f(x)$ as a function of $x$, and use the picture to estimate the area, and you've got an estimate of the energy.  Riemann's definition is obviously not involved in all this.
Riemann's definition gives criteria for when a number is too small or too large to be the integral.  Rather than defining infs and sups, I might say the integral is defined as the only number (if there is only one) that is neither too large nor too small.
(In the same way, Lebesgue's definition, applied to non-negative functions, says the integral is the smallest number that's not too small.  And if every number is too small, then it's $\infty$.  The criterion for when it's too small is simple.  The complicated part is only constructing the measure and showing it's countably additive, etc.)