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Find the volume under the graph of z = f(x, y) = sin(3x) and over the given rectangular region 0 ≤ x ≤ 4, 0 ≤ y ≤ 4. (Use symbolic notation and fractions where needed.) V =
Use Fubini's Theorem to find the double integral defined over the rectangular region R, -1 ≤ x ≤ 2,0 ≤ y ≤ 1. (Use symbolic notation and fractions where needed.) (5x² - 8y) dA= R
Let f(x, y) = 4x(8- y) be defined over the region R shown in the figure. yo 3- 14 (0,3) (0,0) Σf(uk, Uk)A Ak = k=1 (3.3) (a) Find the Riemann sum of f(x, y) over R by partitioning the region into nine congruent subsquares with sides Ax; = 1, i = 1, 2, 3 and Ay; = 1, j = 1, 2, 3. Choose the lower right corner of each subsquare as (uk, Uk), k = 1, 2, 3, ..., 9. (Use symbolic notation and fractions where needed.) (3,0) 9 Σ f(uk, Uk)AAk = k=1 (b) Find the Riemann sum of f(x, y) over the partition used in (a) but choose the upper left corner of each subsquare as (uk, Uk), k = 1, 2, 3, ..., 9. (Use symbolic notation and fractions where needed.)