What does one integral tell for two variable function?

Question

Let's assume I have function:

$ f(x,y) $

Solving double integral: $ \int \int f(x,y)\ dx\ dy $ would give me volume of this body function. For example cubic centimeters.

But what if I use only one integral:

$ \int f(x,y)\ dy$

What do I get out then? A volume, an area? I have hard time to visualize this

Answer 1

You will get a function of x. For example, if you were integrating over some 2D domain, this would be a function yielding the length of the line in the y direction at a fixed x.

Answer 2

$\int f(x,y)dy$ is still a function of $x$. You could define it as $g(x)$ up to a constant of integration. When the double integral gives you a volume it needs to be a definite one. If you use a particular range of $y$ for this integral, $g(x)$ has no arbitrary constant.