Divide R into four squares with m = n = 2, m = n = 2, and choose the sample point as the midpoint of each square: ( 1 / 2, 1 / 2 ), ( 3 / 2, 1 / 2 ), ( 1 / 2, 3 / 2 ), and ( 3 / 2, 3 / 2 ) ( 1 / 2, 1 / 2 ), ( 3 / 2, 1 / 2 ), ( 1 / 2, 3 / 2 ), and ( 3 / 2, 3 / 2 ) to approximate the signed volume.Divide R into four squares with m = n = 2, m = n = 2, and choose the sample point as the upper right corner point of each square ( 1, 1 ), ( 2, 1 ), ( 1, 2 ), ( 1, 1 ), ( 2, 1 ), ( 1, 2 ), and ( 2, 2 ) ( 2, 2 ) ( Figure 5.6) to approximate the signed volume of the solid S that lies above R R and “under” the graph of f.Set up a double integral for finding the value of the signed volume of the solid S that lies above R R and “under” the graph of f.Setting up a Double Integral and Approximating It by Double SumsĬonsider the function z = f ( x, y ) = 3 x 2 − y z = f ( x, y ) = 3 x 2 − y over the rectangular region R = × R = × ( Figure 5.5).