Evaluatex2 + y?- z dS where S is the partSof the cone22x2 + y? that lies between theplanes z = 2 and z = 3.%3D

Answered: x2 + y? – z dS where S is the part S…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Evaluate
x2 + y?
- z dS where S is the part
S
of the cone
22
x2 + y? that lies between the
planes z = 2 and z = 3.
%3D

Expert Solution

Step 1

Given,

$\iint_{s} x^{2} + y^{2} - z d S$

And the cone is $z^{2} = x^{2} + y^{2}$ lies between the planes $z = 2 and z = 3$ .

Now, the parametric equation is,

$r = (z \cos θ, z \sin θ, z)$

$\begin{array}{rcl} J a c o b i a n (J) & = & r_{θ} \times r_{z} \\ J & = & (- z \sin θ, z \cos θ, 0) \times (\cos θ, \sin θ, 1) \\ J & = & |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - z \sin θ & z \cos θ & 0 \\ \cos θ & \sin θ & 1 \end{matrix}| \\ = & |z \cos θ \hat{i} - z \sin θ \hat{j} - z \hat{k}| \\ = & \sqrt{{(z \cos θ)}^{2} + {(- z \sin θ)}^{2} + {(- z)}^{2}} \\ = & z \sqrt{2} \end{array}$

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