First, you need to unlearn what you think you understand. "Area under the curve" is NOT the "meaning" of the integral, it is one possible application. Nor are "surface area" and "volume" the "meanings" of the double and triple integrals. They are specific problems for which the integral can be used. An integral, like any mathematical object, has
no "meaning", as you are using the word, apart from a specific application.
For something like
you
can think of f(x,y) as being the height of some object above each point (x, y) in the plane. In that way the integral is the volume of the object. But you could also have f(x,y) as the
density of a two-dimensional object. Then the integral would be the total mass of the object. You could also have f(x,y) as the pressure of a liquid or gas on a plane. Then the integral gives the total force on that plane.
For
, if f(x,y,z) is the density of the object, then the integral is total mass. Or it could be the charge density so that the integral is the total charge.