(Calculus) Integrating with powers of dx How do you resolve $\int x (dx)^2$?  The notes I am seeing says the term goes to zero, but I don't understand how.  Please help.
 A: Let us interpret $\int_a^b x(dx)^2$ as a limit of Riemann sum. We have
$$\int_a^b x(dx)^2 = \lim_{n \to \infty}\sum_{k=1}^{n}\left(a+k\cdot\dfrac{b-a}n\right)\left(\dfrac{b-a}n\right)^2 = \lim_{n \to \infty} \left(an+\dfrac{n(n+1)(b-a)}{2n}\right)\left(\dfrac{b-a}n\right)^2$$
Hence, we obtain
$$\int_a^b x(dx)^2 = \lim_{n \to \infty} \left(\dfrac{a(b-a)^2}n+\dfrac{(n+1)(b-a)^3}{2n^2}\right) = 0$$
This is true with any higher order power of $(dx)$, i.e., if $\displaystyle \int_a^b f(x) dx \in \mathbb{R}$, then $\displaystyle \int_a^b f(x) (dx)^p$, where $p>1$ is zero.
A: You need to assign a meaning to your intended calculation. 
What does $x (dx)^2$ mean in the first place? 
Is it the same as $x\, dx \,dx$?
What kind of integration is exactly meant if you add an summation $\int$ symbol to your term? What input mathematical objects what output objects what summation procedure?
There is no useful integration for this kind of term defined that I am aware of (maybe some non-standard analysis folk has one).
