I was looking at the equation for the differential force $dF$ caused from a stress tensor $\sigma$ acting on a differential surface $dS$. In mathematical terms,
$dF=\sigma \cdot dS$
Taking the integral of both sides yields.
$$F=\int \sigma \cdot dS$$
What is the nature of this integral? How does on go about calculating it? I would like to see an example given a non-trivial $\sigma$.
Can this be generalized to $F=\int \sigma \cdot dV$?
My idea was to integrate with respect to the variables $x$ and $y$ and represent $dS$ as a normal vector multiplied by the $dx$ and $dy$ differentials. Taking the product of the right hand side, you then integrate with respect to the differentials.