   Chapter 16.7, Problem 2E

Chapter
Section
Textbook Problem

A surface S consists of the cylinder x2 + y2 = 1, −1 ⩽ z ⩽ 1, together with its top and bottom disks. Suppose you know that f is a continuous function withf(±1, 0, 0) = 2 f(0, ±1, 0) = 3 f(0, 0, ±1) = 4Estimate the value of ∬S f(x, y, z) dS by using a Riemann sum, taking the patches Sij to be four quarter-cylinders and the top and bottom disks.

To determine

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

Explanation

Given data:

Surface S consists of the cylinder x2+y2=1 , 1z1 .

f(±1,0,0)=2 , f(0,±1,0)=3 and f(0,0,±1)=4 .

Each quarter-cylinder has surface area is 14[2π(1)(2)]=π , and the top and bottom disks have surface area π(1)2=π .

Consider that (0,0,1) has a sample point in the top disk, (0,0,1) has a sample point in the bottom disk and (±1,0,0),(0,±1,0) in the quarter-cylinders.

Find Sf(x,y,z)dS .

Sf(x,y,z)dS{[f(1,0,0)](π)+[f(1,0,0)](π)+[f(0,1,0)](π)+[f

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