Can someone provide an example of Lebesgue integration on ordinary functions? I'm trying to understand Lebesgue integral and integration but I'm completely mired by many author's motivation of this subject as only a tool for pathological functions such as the Dirichlet pathological function. I entirely appreciate the functionality of Lebesgue integration for this purpose but there just seems to be a huge gap between the classical calculus that I've taught since high school and this concept.
Can someone show how Lebesgue integration becomes Riemann integration for ordinary/smooth functions such as a constant or a polynomial?
Thank you

When we want to do Lebesgue integration to a Riemann integrable function like the ones you have described, it is helpful to recall Riemann sums. 
For simplicity let $f$ be nonnegative, and let us integrate over a finite, closed interval $[a,b] \subset \mathbb R$.
We get the Lebesgue integral as the supremum of the integral of simple functions less than $f$.
Let us look at lower Riemann sums; for a partition $P = \{x_i | a < x_0, x_i < x_{i+1}, x_n < b \}$, the lower Riemann sum is the sum of rectangles "just barely" below the function on these intervals. This is the integral of a simple function, 
with value $\min\{f(x),x \in [x_i,x_{i+1}]\}$ on the interval $[x_i,x_{i+1}]$- so it is included in the supremum for the Lebesgue integral.
So for a riemann integrable function, as all the lower Riemann sums are integrals of simple functions, we can see that the Lebesgue Integral (the supremum of integrals of simple functions less than $f$) is greater than the Riemann integral (the supremum of lower riemann sums). From here it is not hard to show they are both bounded above by the upper Riemann integral.
For this simple case, Lebesgue is just a generalization of Riemann.
