Please help me Evaluate |/ ax² + y? – z dS where S is the-S22= x + y' that liespart of the conebetween the planes z = 1 and z =%3D5.

Name: Please help me Evaluate |/ ax² + y? – z dS where S is the-S22= x + y'
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Description: Please help me Evaluate |/ ax² + y? – z dS where S is the-S22= x + y' that liespart of the conebetween the planes z = 1 and z =%3D5.

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

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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Please help me

Evaluate |/ a
x² + y? – z dS where S is the
-
S
22
= x + y' that lies
part of the cone
between the planes z = 1 and z =
%3D
5.

Expert Solution

Step 1

Given,

$f (x, y, z) = x^{2} + y^{2} - z$

Here, $S$ is the part of the cone $z^{2} = x^{2} + y^{2}$ that lies between the planes $z = 1$ and $z = 5$

Parametrizing the given surface,

$P (r, θ) = (r \cos θ, r \sin θ, r)$

Then,

$\begin{array}{rcl} P_{r} & = & (\cos θ, \sin θ, 1) \\ P_{θ} & = & (- r \sin θ, r \cos θ, 0) \\ P_{r} \times P_{θ} & = & (- r \cos θ, r \sin θ, r) \\ {||P_{r} \times P_{θ}||}^{2} & = & r^{2} \cos^{2} θ + r^{2} \sin^{2} θ + r^{2} \\ = & 2 r^{2} \\ ||P_{r} \times P_{θ}|| & = & \sqrt{2} r \end{array}$

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