Double integral definition

I don't understand the definition of double integral.
For instance in the functions with single variable the definite integral was defined as Riemannian sum as: $$\lim_{n\to\infty}\sum_{k=1}^{n} f(c_k)\delta x_k$$ In which I could assume $f(c_k)$ as the height and $\delta x_k$ as the width of the rectangles we're going to calculate the sum of but for the double integration in the region $R$ we've got the definition:$$\lim_{\delta A\to\infty}\sum_{k=1}^{n}f(x_k,y_k)\delta A_k $$Now I can assume the $\delta A$ to be the surface of small rectangles which cover the region $R$.Then what $f(x_k,y_k)$ supposed to be?I'm trying to learn by analogy that $f(c_k)$ was assumed to be the height but now what is the $f(x_k,y_k)$ supposed to be in the double integral definition?
Thanks in advance
P.S: Notice that I'm trying to find out what $f(x_k,y_k)$ tries to represent when we're trying to find the surface of region $R$ not the volume!