Double Integral and Triple Integral question

So I understand the meaning of an integral of a function, which is the area of the region under the curve that represents that function within the limits of integration.

I also understand the meaning of $\displaystyle \iint_D dxdy$ which is the area of the region D and $\displaystyle \iiint_V dxdydz$ which is the volume of a solid V.

What I don't understand, even though I do it all the time, is the meaning of a double integral (and a triple integral) of a function. For example: $\displaystyle \iint_D (x^2+\sin y)dxdy$ and $\displaystyle \iiint (x^2+\sin y \cos z)dxdydz$.

Thanks in advance.