Question

(1 point) Calculate the Riemann sum for f(x, y) = 3.1 - 6y and domain D in Figure 2 with two choices of sample points, and o. Which do you think is a better approximation to the integral of f over D? Why? 4 o 3 O o O! Oo o o o 0 1 FIGURE 2 $34) $34) = 0

Answer 1

Given f(x, y) = 3x - 64 NOVO The rectangle have side length of AXc - Ay = 1 ( a) sample points (0) The points are ! - (1, 1) (2, 1) (3,1) (1,2) (212) (3, 2) slobode (1,3) (2,3) (3,3) C+2+7+2.- (1,4) (2,4) (3,4) f = 3xc-by Осо substitute all points in f, we get! = -3 to +3 +(-9)+(-6)+(-3) + (-15) + (-12) +(-9) + (-21) + (-18) + (-15) = - 108 TO this, multiply Ax and by, we get -108 x1x1 -108 S. 3,4 () = -108

b) sample points (0) :- using These (0.5, 0.5) (1.5, 0.5) (2.5, 0.5) (0.5, 1.5) (1.5,1.5) ch (2.5, 1.5) (o.5 12.5) (1.5, 2.5) (2-8,205) (0.5,3.5) (1.5,3.5) (2.5, 3.5) - Substitute in 3x by we get -1.5+ 1.5 +4.5+ (-7.5) + (-4.5) + (-15) + (-13-5) +(105) + (-7.5) + (-19.5)+(-16-5) + (-13-5) 11 - 90 and here. want to multiply ax and Ayo ton - 90x1x1 3-90 33,4 (0) = -90 The sample point (o) is much more accurate because it is not an overestimated (or) understimate points taken were the midpoints of each little region. because the