EXAMPLE 1 Estimate the volume of the solid that lies above the square R = [0, 2] x [0, 2] and below the elliptical paraboloid z = 27 - x2 - 5y². Divide R into 4 equal squares and choose the sample point to be the upper right corner of each square Rj. Sketch the solid and the approximating rectangular boxes. SOLUTION The squares are shown in the top figure. The paraboloid is the graph of f(x, y) = 27 – x - 5y? and the area of each square is ΔΑ - Approximating the volume of the Riemann sum with m = n = 2, we have 2 i = 1j = 1 f(1, 1)AA + f(1, 2)AA + f(2, 1)AA + f(2, 2)AA = 21 + 6(1) + (1) + 3(1) This is the volume of the approximating rectangular boxes shown in the bottom figure.

EXAMPLE 1 Estimate the volume of the solid that lies above the square R = [0, 2] x [0, 2] and below the elliptical paraboloid z = 27 - x2 - 5y². Divide R into 4 equal squares and choose the sample point to be the upper right corner of each square Rj. Sketch the solid and the approximating rectangular boxes. SOLUTION The squares are shown in the top figure. The paraboloid is the graph of f(x, y) = 27 – x - 5y? and the area of each square is ΔΑ - Approximating the volume of the Riemann sum with m = n = 2, we have 2 i = 1j = 1 f(1, 1)AA + f(1, 2)AA + f(2, 1)AA + f(2, 2)AA = 21 + 6(1) + (1) + 3(1) This is the volume of the approximating rectangular boxes shown in the bottom figure.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.4: Linear Programming

Problem 4E

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