3 2 We can estimate the volume of the given solid using t1e Riemann sum 2 2 f(x,., y].)AA: with f(X, y) = xy. i= 1]: 1 Since 7 s x s 13 and m : 3, then Ax : I J \"E , and since 3 s y s 7 and n : 2, then Ay : I HE . Consequently, we have AA=I HE. Step 2 When i: 1 and ] : 1, we are referring to the coordinates of the rectangle in the lower left corner of the grid. We are told to use the upper right corner as a sample point. Therefore, for i: 1 andj : 1, the sample point would be (x, y) : ( - J 9' 5 ). Step 3 Similarly, for example, when i = 2 andj = 1, the sample point is (X, y) = (I I: 11 , 5), and when when i = 3 andj = 2, the sample point is Calculating the other sample points, we can therefore write the volume as 3 2 V: Z Z fog ,ijA : AA(f(9, 5) + f(11, 5) + f(13, 5) + f(9, 7) + f(11, 7) + f(13, 7)). i= 1} = 1 Factoring out AA = 4, substituting the values of the function at the given sample points, and evaluating, we can conclude that the volume is estimated by Vz(4)(45+55+65+63+77+ ) = (4)( )