Let S₁ be the hemisphere with Cartesian equation z = 6 - 36 – x² - y² and S₂ be the upper nappe of the cone x² + y² - 3z² = 0. a. The curve of intersection of S₁ and S₂ is a circle. Find the radius and the coordinates of the center of this circle. b. Find an equation in spherical coordinates for S₁. c. Let G₂ the solid enclosed by S₁ and S₂. a. Set up an iterated triple integral in cylindrical coordinates that yields the volume of G₂. Do not evaluate the integral. b. Find the volume of G₂ using a triple integral in spherical coordinates.
Let S₁ be the hemisphere with Cartesian equation z = 6 - 36 – x² - y² and S₂ be the upper nappe of the cone x² + y² - 3z² = 0. a. The curve of intersection of S₁ and S₂ is a circle. Find the radius and the coordinates of the center of this circle. b. Find an equation in spherical coordinates for S₁. c. Let G₂ the solid enclosed by S₁ and S₂. a. Set up an iterated triple integral in cylindrical coordinates that yields the volume of G₂. Do not evaluate the integral. b. Find the volume of G₂ using a triple integral in spherical coordinates.
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:Let S₁ be the hemisphere with Cartesian equation z = 6 – √√/36 – x² - y² and S₂ be the
upper nappe of the cone x² + y² − 3z² = 0.
a. The curve of intersection of S₁ and S₂ is a circle. Find the radius and the coordinates of
the center of this circle.
b. Find an equation in spherical coordinates for S₁.
c. Let G₂ the solid enclosed by S₁ and S₂.
a. Set up an iterated triple integral in cylindrical coordinates that yields the volume of
G₂. Do not evaluate the integral.
b. Find the volume of G₂ using a triple integral in spherical coordinates.
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