Consider the torus as $T^2 = \mathbb R^2/ \mathbb Z^2$, let $A: = \{7x+3y = 0\} \subset \mathbb R^2$. Its image, denoted by $\pi(A)$, in $T^2$ is a one dimensional submanifold. Now I want to calculate $\int_{\pi(A）} dx$.
The image of $\pi(A)$ are several parallel lines inside the square, the integration of $dx$ is just the sum of their length in the $x$ direction. But, still this does not seem to be easy to compute. What is the standard way of doing this?
Parametrize it by $x$. We have the parametrization $\gamma(x) = (x,-7/3x)$, for $x\in [0,3]$.
Then $$\int_{\pi(A)}dx = \int_0^3 d(x\circ \gamma) = \int_0^3 dx = 3.$$
Since you haven't specified the orientation, the answer could also be $-3$.
This should make sense intuitively, since the line wraps around the torus in the $x$ direction 3 times. (If you graph $7x+3y=0$, or $y = \frac{-7}{3}x$, the first positive $x$-value at which the graph passes through a lattice point is $x=3$)