Intuitively, we think of _A R f(x,y)dA as measuring the volume between a surface and the xy-plane:

Recall from Calc 1:
To approximate the area under a positive-valued function y=f(x) over the interval I= [a,b], we

• partition the interval [a,b] into n smaller subintervals. The ith subinterval will have length &Delta x_i; usually we partition so that all subintervals have length &Delta x.
• pick evaluation points, one from each subinterval. Call these c_i.
• Add up the areas of the rectangles that have as their base &Delta x_i, and as their height f(c_i).
  
  
• Take the limit as the number of rectangles approaches infinity.

Extend this idea to functions of two variables, f(x,y):
We want to approximate the volume under the positive-valued function z=f(x,y) over the rectangle R=[a,b] x [c,d]:

• partition the rectangle [a,b] x [c,d]: into n rectangles, by partitioning both [a,b] and [c,d]. Usually we'll partition both [a,b] and [c,d] into k subintervals, giving k^2 rectangles, each of area &Delta A.
• pick evaluation points, one from each subinterval. Call these points (u_i, v_i).
• Add up the values of the boxes that have as their base area &Delta A and as their height f(u_i,v_i).
• Take the limit as the number of boxes approaches infinity (specifically, as the diagonal of the largest box approaches zero).