Answered: Calculate s f(x,y, z) dS, where S is…

Calculate s f(x,y, z) dS, where S is the part of the plane x + y + z = 4, where x, y, z > 0, and f(x, y, z) = z. (Use symbolic notation and fractions where needed.) f(x, y, z) dS =

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.3: Lines And Planes

Problem 44EQ

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Question

Calculate fs f(x, y, z) dS, where S is the part of the plane x + y + z = 4, where x, y, z > 0, and f(x, y, z) = z.
(Use symbolic notation and fractions where needed.)
/| S(x, y, z) dS =

Expert Solution

Step 1

We have to calculate $\iint_{S} f (x, y, z) d S$ where, $S$ is the part of the plane $x + y + z = 4$ where, $x, y, z \geq 0$

And $f (x, y, z) = z$ .

Now, solve $x + y + z = 4$ for $z$ , we get

$z = 4 - x - y$

Now,

$\begin{matrix} d S = \sqrt{1 + {(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2}} d A \\ = \sqrt{1 + {(- 1)}^{2} + {(- 1)}^{2}} d A \\ = \sqrt{1 + 1 + 1} d A \\ = \sqrt{3} d A \end{matrix}$

Hence, $d S = \sqrt{3} d A$ .

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