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Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of f(x, y) = 5 + xy above the rectangle R with 0 < x < 2, 0< y< 5. upper bound = lower bound %3D 0, x = 0.5, Notice that R will be partitioned into subrectangles with the lines x = x =, x =, and x = 2 and the lines y = 0, y = 1.25, y =, y =, and y = 5. Then you will have 16 subrectangles, each of which we denote Rab, where (a, b) is

Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of f(x, y) = 5 + xy above the rectangle R with 0 < x < 2, 0< y< 5. upper bound = lower bound %3D 0, x = 0.5, Notice that R will be partitioned into subrectangles with the lines x = x =, x =, and x = 2 and the lines y = 0, y = 1.25, y =, y =, and y = 5. Then you will have 16 subrectangles, each of which we denote Rab, where (a, b) is

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of f(x, y) = 5+ xy above the 0 < y< 5. rectangle R with 0 < x < 2, upper bound lower bound = Notice that R will be partitioned into subrectangles with the lines x = 0, x = 0.5, x =, x =, and x = 2 and the lines y 0, y = 1.25, y =, y =, and y = 5. Then you will have 16 subrectangles, each of which we denote Ra.b) , where (a, b) is ne location of the lower-left corner of the subrectangle. You want to nd a ower bound and an upper bound for the volume above each subrectangle. The lower bound for the volume of R(a.b) is (Min of f on R(a,b)) because the area of Ra.b) is 0.5 · 1.25 =. The function f(x, y) = 5 + xy increases with both x and y over the whole region R. Thus, Min of f on R(a,b) = f(a, b) =, because the minimum on each subrectangle is at the corner closest to the origin. The calculation for the upper bound is similar replacing the smallest value of f by the largest value.
Transcribed Image Text:Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of f(x, y) = 5+ xy above the 0 < y< 5. rectangle R with 0 < x < 2, upper bound lower bound = Notice that R will be partitioned into subrectangles with the lines x = 0, x = 0.5, x =, x =, and x = 2 and the lines y 0, y = 1.25, y =, y =, and y = 5. Then you will have 16 subrectangles, each of which we denote Ra.b) , where (a, b) is ne location of the lower-left corner of the subrectangle. You want to nd a ower bound and an upper bound for the volume above each subrectangle. The lower bound for the volume of R(a.b) is (Min of f on R(a,b)) because the area of Ra.b) is 0.5 · 1.25 =. The function f(x, y) = 5 + xy increases with both x and y over the whole region R. Thus, Min of f on R(a,b) = f(a, b) =, because the minimum on each subrectangle is at the corner closest to the origin. The calculation for the upper bound is similar replacing the smallest value of f by the largest value.
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