We know that in the theory of distributions $\int_{-1}^1 \delta(x)dx=1$.
Now consider a function $f(x)$ which is equal to $1$ at $x=0$ but otherwise is zero.
Did anyone propose a generalization of integration, under which $\int_{-1}^1 f(x)dx$ is an infinitesimal?
This would allow to compare infinitesimals, multiply them by real numbers, etc...
Well this would have an exact value as $dx$ when $dx$ is considered the small number you break Reimann integrals up into, so I think it should have the same value as the infinitesimal typically denoted $\epsilon$ or $\frac{1}{\omega}$ that is used in non-standard analysis (analysis that makes infinitesimals rigorous).