4. Find the surface area of the circular cone S = {(x, y, z) € R³ | 1² + y² − z² = 0, z ≥ 0, x² + y² < 1} using the following techniques. (i) Using a spherical double integral. You must begin by redefining the cone in spherical coordi- nates. (ii) Using the projection technique Hints: dA dA=. A with projection orientation î* = Î. (a) The projection surface S* should only permit variations in z and y but ñ- * will result in an expression containing all three variables. You should use the definition of S to remove z. (b) Once you have completed the operation in (a), the integral over S* should be very simple. You may either recast the integral into an appropriate coordinate system to complete the evaluation or you may simply state its value with an appropriate description / explanation of the surface integral that remains.
4. Find the surface area of the circular cone S = {(x, y, z) € R³ | 1² + y² − z² = 0, z ≥ 0, x² + y² < 1} using the following techniques. (i) Using a spherical double integral. You must begin by redefining the cone in spherical coordi- nates. (ii) Using the projection technique Hints: dA dA=. A with projection orientation î* = Î. (a) The projection surface S* should only permit variations in z and y but ñ- * will result in an expression containing all three variables. You should use the definition of S to remove z. (b) Once you have completed the operation in (a), the integral over S* should be very simple. You may either recast the integral into an appropriate coordinate system to complete the evaluation or you may simply state its value with an appropriate description / explanation of the surface integral that remains.
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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Transcribed Image Text:4. Find the surface area of the circular cone S = {(x, y, z) € R³ | x² + y² − z² = 0, z ≥ 0, z² + y² ≤ 1}
using the following techniques.
(i) Using a spherical double integral. You must begin by redefining the cone in spherical coordi-
nates.
(ii) Using the projection technique [dA= [₁.1
Hints:
dA
|ñ.ñ |
with projection orientation * = k.
(a) The projection surface S* should only permit variations in x and y but ñ-ñ* will result in
an expression containing all three variables. You should use the definition of S to remove z.
(b) Once you have completed the operation in (a), the integral over S* should be very simple.
You may either recast the integral into an appropriate coordinate system to complete the
evaluation or you may simply state its value with an appropriate description / explanation
of the surface integral that remains.
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