Question

How does a double integral gives area ? I cannot imagine it

Answer 1

In general, double integrals calculate volume (or can be interpreted as such).

That is:

$\displaystyle \iint_{A} f(x,t)dA$ 

gives the net volume under the function $f(x,y)$ and the x-axis.

However, if $f(x,y)=1$, then the height of the solid is 1 and its volume is equivalent to its base area.