Estimate the volume of the solid that lies below the surface z=xy and above the rectangle r. use a riemann sum with m=3, n=2, and take the sample point to be the upper right corner of each square.
r = {(x, y): 0< =x< =6, 0< =y< =4}

(a) Using upper right endpoints for each of the little 'grids' yields (with f(x,y) = xy) 
∫∫R xy dA 
≈ 2 * 2 [f(2, 10) + f(4, 10) + f(6, 10) + f(2, 12) + f(4, 12) + f(6, 12)] 
= 4 [20 + 40 + 60 + 24 + 48 + 72] 
= 1056. 

(b) Using the midpoint for each of the little 'grids' yields 
∫∫R xy dA 
≈ 2 * 2 [f(1, 9) + f(3, 9) + f(5, 9) + f(1, 11) + f(3, 11) + f(5, 11)] 
= 4 [9 + 27 + 45 + 11 + 33 + 55] 
= 720.