University:
Seattle Pacific University
Course:
MAT 1236 | Calculus III
Academic year:

2024
Views:

198

Pages:

1
Author:

Nomes67i

Workshop #19 1. If we want to integrate over a rectangle R = (x, y) a ≤ x ≤ b, c ≤ y ≤ d = [a, b] × [c, d], then we ZZ m X n X ∗ )∆A . To estimate this integral, we can f (x, y)dA = lim f (x∗ij , yij have the definition ij m,n→∞ ZZ f (x, y)dA ≈ use R m X n X i=1 j=1 ∗ )∆A f (x∗ij , yij ij i=1 j=1 R We want to use this to estimate the volume of the solid that lies below the surface z = 4xy and above the rectangle R = [2, 8] × [4, 6] by using a Riemann sum and the 3-by-2 grid below (then m = 3 and n = 2 ). Choose the points to evaluate the heights to be the lower left corner. ∗ )= (x∗11 , y11 ∗ ∗ )= (x12 , y12 ∗ )= (x∗21 , y21 ∗ ∗ )= (x22 , y22 ∗ )= ∗ (x31 , y31 ∗ )= (x∗32 , y32 Using the appropriate definitions and proper notation, estimate this volume. 2. Integrate completely. Z π/2 Z π (a) (xy + sin x)dxdy 0 0 Z 2Z (b) 0 ZZ (c) 1 ex+y dxdy 0 xex+y dA, where R = [0, 2] × [0, 3] R ZZ (d) R xy dA, where R = [0, 1] × [0, 4] 1+x