Lebesgue Integral, Riemann Integral and Integrals of all sorts I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I will be a graduate student in the fall and I still don't know much about this integral. Come someone shed some light on when might one want to use the Lebesgue integral versus the Riemann integral. Please humor me with the simplest of examples. Can integrals which a solved with elementary techniques on the Riemann integral also be solved with the Lebesgue integral? Is there any geometric interpretation of the Lebesgue integral (the interpretation of the Riemann integral is usually length, area, volume, etc as we extend to higher dimensions). Also, are there any other integrals? And why aren't they introduced in the usual undergraduate setting?
 A: *

*Yes there are other functions which are Lebesgue integrable but not Riemann integrable. The most common example is the indicator of the rationals 
$$
  I_\mathbb{Q}(x)
= \begin{cases}
  1 & x \in \mathbb{Q} \\
  0 & x \notin \mathbb{Q} \\
  \end{cases}.
$$
Here the lower and upper Riemann sums will always be $0$ and $1$, and hence the Riemann integral won't exists, but the Lebesgue integral exists and gives the value $0$ that you might have expected.

*Yes, you can use the tools you learned for computing Riemann integrals to compute Lebesgue integrals, so long as the function is Riemann integrable. This is because the integrals agree when both exist.

*In terms of a geometric interpretation, one way to look at it is as follows. Both integrals compute the volume under surfaces, or area under curves corresponding to "nice" functions for which you can think of these regions as areas or volumes. But the distinction is that Riemann's integral does it by adding up volumes of vertical rectangles of equal width arranged so that they approximate the region (the domain is partitioned), whereas Lebesgue's integral does it by adding up volumes of horizontal rectangles of equal height arranged so that they approximate the region. In both cases the assumption of equal height/width can be dropped but you get the idea.

*Yes, there are other integrals, such as the Riemann-Stieltjes and the Ito. I cannot say why some integrals are taught at certain points and others are not, but I do agree with the
standard pedagogical approach since my feeling is that Riemann's integral is the simplest, and gets the main idea across without the need for too much theory, yet is quite robust and has many applications. 

A: I cannot comment yet, so I'll post this here.
There are many reasons to use the Lebesgue integral and to learn it. I'm going to concentrate on a practical aspect you are probably familiar with.
It turns out many desirable equalities essentially reduce to checking whether integration and limits commute. To list some examples you may have already encountered, one often wonders about $\displaystyle{\sum_{n\in \mathbb{N}}}\int f_ndx\overset{?}{=}\int(\displaystyle{\sum_{n\in\mathbb{N}}f_n)dx}$, and $\frac{d}{dt}\displaystyle{\int}f(x,t)dx\overset{?}{=}\displaystyle{\int}\frac{\partial f(x,t)}{\partial t}dx$. To see where the interchange of limit and integral hides, replace $\sum_{n\in\mathbb{N}}$ by its definition, and the dervative with its definition. If the integrals are taken to be Riemann integrals, stringent conditions are needed to guarantee equality (unioform convergence of the sum, for instance). Furthermore, the limit of Riemann integrable functions need not itself be Riemann integrable, so perhaps the expression $\displaystyle{\int(\displaystyle{\sum_{n\in\mathbb{N}}f_n)dx}}$ does not even have any meaning!
Lebesgue integrability has the nice property of being closed under limits, so first of all, we needn't worry about the meaning of expressions like $\displaystyle{\int(\displaystyle{\sum_{n\in\mathbb{N}}f_n)dx}}$. Second, there are very powerful and intuitive convergence theorems for the Lebesgue integral which give mild and natural conditions under which integration and limits commute (by natural and intuitive I mean analogous to conditions on sequences and series of numbers).
The geometric interpretation of the Lebesgue integral is in a sense more natural, as its theoretical context generalizes the notions of length, area, and volume. Imagine the function which takes the value $c$ on the intevral $[a,b]$ and $0$ otherwise. By definition, the Lebesgue integral of this function $f$ is the measure (length) of the interval, times the value $f$ takes on it. Suppose now that $f$ were $0$ outside a set $A$ which is not an interval, but rather a stranger set made of intervals and points. The integral of this function would still be the measure of $A$ (its generalized length) times $c$. Basically, the Lebesgue integral abstracts from literally taking the integral to be the area under the curve.
In one of the comments, the terrific book A Garden of Integrals by Burke was mentioned. Another nice book exploring different integrals is The Integrals of Lebesgue, Denjoy, Perron, and Henstock, by Gordon.
