Q:

Vanishing of the integration along vertical line

2- Show that if $C$ is a vertical line segment $c \leq y \leq d,$ and if $F$ is a function of 2 variables defined on $C$, then $$\int_{C} F(x,y)dx = 0$$.
I understand that the integration is calculating the area under the curve and because we are integrating on a vertical line so the area is zero. is this the meaning of the question ?
Also I am stucked in the proof as all the theorems and definitions in this section include parametrizations, or shall I use this question Proving a Line-integration along a parametrized curve identitiy.  in proving it?if so how?.  Any help in the proof will be appreciated.

A:

Yes the area under this curve will be zero. You can do this a bit more rigorously by consider the following limiting process. Consider a line defined by
$$y(x) = c + \frac{x}{\epsilon}(d-c), \quad x\in [0,\epsilon]$$
so that $y(0) = c$ and $y(\epsilon) = d$ and consider
$$|\int_0^{\epsilon} F(x,y(x))dx| \leq \max_{(x,y)\in [0,\epsilon]\times[c,d]} |F(x,y)|\int_0^{\epsilon} dx \leq C\epsilon \rightarrow 0$$
This is just taking the line connecting $(0,c)$ and $(\epsilon,d)$, so in the limit that $\epsilon \rightarrow 0$, this approaches the appropriate segment at $x=0$. This holds for any $x$ so the integral must be zero just as the area under the curve is zero. 

