I'm trying to find the flux of the vector field $$\textbf{v}(x,y)=x^{2}\textbf{i}-2xy\textbf{j}$$ through the «stream pipe»* given by the lines $y=0$ and $y=\frac{1}{x^{2}}$, as shown in this figure.
My first thought was to parameterize the lines. However, the parametrization $\textbf{r}(t)=t\textbf{i}+1/t^{2}\textbf{j}$ for $-\infty<t<\infty$ of the blue line is undefined at zero, so the parametrization would not be everywhere defined, and to me that looks like a problem.
I tried using Gauss', but I don't see how to define the volume over which to integrate.