Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$
I ask this question because of the generalization to multiple dimensions of integration by parts over a region $\Omega$, which includes a line integral $$\oint_{\partial \Omega} \nabla f \cdot \vec n,$$ where $\partial \Omega$ is the boundary of $\Omega$. In $\mathbb{R}^1$, the boundary of $\Omega = [a,b]$ is the finite set $\partial \Omega = \lbrace a, b \rbrace$, and the normal vectors at $a$ and $b$ are $-1$ and $1$, respectively, so the integral simplifies to just $f(b) - f(a)$.
This makes sense intuitively. Does the math work, or is it just a helpful explanation?
Depends on how you are defining integration. If it's the usual sense of Riemann, you just get 0. If you're using point mass measures $\delta_n$, then "yes" it's a sum. You can read about such things here.