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!
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$f(x_k,y_k)$ represents the height of the right rectangular prism (ie. the box) of base area $\delta A_k$. The sum of the volume of all of these boxes will approximate the volume under the surface $z=f(x,y)$ and above $R$.
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$\begingroup$ You're trying to calculate volume, what if we try to calculate the surface?then what would the f(x,y) represent? $\endgroup$– FreeMindMay 3, 2014 at 21:14
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$\begingroup$ @MrWho: It could also represent the density of the flat material at that point. Then the integral would represent the total mass of the material. $\endgroup$– JaredMay 3, 2014 at 22:48