   Chapter 16.7, Problem 3E

Chapter
Section
Textbook Problem

Let H be the hemisphere x2 + y2 + z2 = 50, z ⩾ 0, and suppose f is a continuous function with f(3, 4, 5) = 7, f(3, −4, 5) = 8, f(−3, 4, 5) = 9, and f(−3, −4, 5) = 12. By dividing H into four patches, estimate the value of ∬H f(x, y, z) dS.

To determine

To find: The value of Hf(x,y,z)dS .

Explanation

Given data:

H be the hemisphere x2+y2+z2=50 , z0 .

f(3,4,5)=7 , f(3,4,5)=8 , f(3,4,5)=9 and f(3,4,5)=12 .

The xz- and yz- planes are used to divide hemisphere (H) into four patches of equal size, each has a surface area equal to 18 the surface area of a sphere with radius 50 .

Therefore,

ΔS=18(4)π(50)2=12π(50)=25π

The sample points in the four patches are (±3,±4,5) .

Find Hf(x,y,z)dS .

Hf(x,y,z)dS{[f(3,4,5)](25π)+[f(3,4,

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