What can be the geometrical meaning of $$ \iint_R dA ~~~~~~~~~~ (1)$$ ? It is the particular case of $$\iint_R {f(x,y)} dA$$ where $f(x,y) = 1 $. What I have found from my search is that (1) represents the area of the region R which is quite intuitive : if we have an area element which is a function of two variables then we must have to integrate it twice to get the area of the region. But if we see it from a different view point then it is the volume under the plane $z=1 $ and above the region R. Well, we can even extend this to triple integrals where $$ \iiint_S dV $$ represents the volume. The thing which is causing the doubt is the substitution of $ f(x,y)$ or $f(x,y,z)$ with $1$ .
So, which interpretation is correct? Am I wrong somewhere in the very essence?
Any help will be much appreciated.
Interpretation is often relative in mathematics. There is no correct interpretation (I don't mean that there may not be inapt interpretations). Often indeed, we have to change the way we look at the same object in order to understand it properly or be able to use it effectively -- even sometimes in the course of solving the same problem. This is what makes things convenient -- viewing things from different perspectives. So, yes, we may view the $$\int_R\mathrm d A$$ either as the area of $R$ or as the volume of a cylinder of unit height and cross section $\partial R.$ What does it matter? It only helps us have more handles on our objects. Indeed, the same integral can be interpreted as many more things. The same happens with other objects across mathematics -- e.g., second order differential equations describe many different phenomena. It is what physicist Eugene Wigner once wondered about -- why mathematics seems to crop up everywhere in nature in such a way as to unite many apparently different phenomena. We don't even need to go to two dimensions. The integral $\int_a^b\mathrm d x$ may be taken to mean the length of the interval $[a,b],$ or the area of the rectangular region $[a,b]×[a,1].$ This type of duality abounds everywhere in mathematics, and it helps in solving more problems in a more unified way. Another is how we may view a real tuple both as a euclidean vector or a point, etc.
So there's no worry. Be flexible. You need not fix interpretations in your mind. Indeed, be open to as many (apt) interpretations of an object as you can get -- it will only improve your intuitive grasp of if, and help you to see various applications and connexions. Good luck!