Intro:
A continuous, differentiable function $F(x,y)$ can be pictured as a surface over the $(x,y)$ plane. At each point in space, we could plot another function $\frac{\partial F}{\partial y}(x,y)$, which would give us another surface.
By integrating this derivative in $y$ direction, we get changes in height to the actual function $F$ due to motion in the $y$ direction. However, what do we get by integrating this derivative in the $x$ direction, and why?
Part I Understand:
Say at some point $(x_0,y_0)$, we hold $x=x_0$, and compute the integral: $$\int_{y_0}^{y_1}{\frac{\partial F}{\partial y}(x,y)\Bigr\rvert_{x = x_0}dy}$$
This would give us an area under a slice of the graph of $\frac{\partial F}{\partial y}(x,y)$, which would correspond to a change in height in the graph of $F(x,y)$ due to moving solely in the $y$ direction.
Part I'm Having Trouble Understanding:
Say that starting at some point $(x_0,y_0)$, we hold $y=y_0$ constant, and compute the integral: $$\int_{x_0}^{x_1}{\frac{\partial F}{\partial y}(x,y)\Bigr\rvert_{y = y_0}dx}$$
This would give us an area under a slice of the graph of $\frac{\partial F}{\partial y}(x,y)$, but an area of a slice taken in the $x$ direction. I'm told that this would correspond to the rate of change of the area given by...
$$\int_{x_0}^{x_1}{F(x,y)\Bigr\rvert_{y = y_0}dx}$$
...with respect to the $y$ direction. However, I'm having trouble intuitively understanding this...or picturing it. Any and all help would be greatly appreciated!
Thanks!