A) estimate the volume of the solid that lies below the surface z = 7x + 5y2 and above the rectangle r = [0, 2]⨯[0, 4]. use a riemann sum with m = n = 2 and choose the sample points to be lower right corners.

In the direction we consider the subintervals [0, 1] and [1, 2] (each with length 1), while in the direction we consider the subintervals [0, 2] and [2, 4] (with length 2). Then the lower right corners of the cells in the partition of are (1, 0), (2, 0), (1, 2), (2, 2).

Let . The volume of the solid is approximately

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More generally, the lower-right-corner Riemann sum over and subintervals would be

Then taking the limits as and leaves us with an exact volume of .