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Find the volume of the solid that lies under the paraboloid == x² + y²₁ above the XY-plane, and inside the cylinder x² + y² 21.

Find the volume of the solid that lies under the paraboloid == x² + y²₁ above the XY-plane, and inside the cylinder x² + y² 21.

Advanced Engineering Mathematics
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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EXAMPLE 4 Find the volume of the solid that lies under the paraboloid = = x² + y²₁ above the XY-plane, and inside the cylinder x² + y² 216. SOLUTION The solid lies above the disk D whose boundary circle has the equation x² + y² = (x-1)² + y² = 1, after completing the square. 2x₂ or In polar coordinates, this circle is 0 (x-1)² + y² = 1 (or r = 2 cos 0) D 2 4 2 Therefore D = {(1,0) | -1 = 0 = 1, r = 2 c05 0} 2 ·2 LOS V = {{{[®c® £4Ð » (H™ • SS₂ ( x² + y²) dA r²r dr do 0 S³ cos" o do $ = 20 cos cos 40 do
Transcribed Image Text:EXAMPLE 4 Find the volume of the solid that lies under the paraboloid = = x² + y²₁ above the XY-plane, and inside the cylinder x² + y² 216. SOLUTION The solid lies above the disk D whose boundary circle has the equation x² + y² = (x-1)² + y² = 1, after completing the square. 2x₂ or In polar coordinates, this circle is 0 (x-1)² + y² = 1 (or r = 2 cos 0) D 2 4 2 Therefore D = {(1,0) | -1 = 0 = 1, r = 2 c05 0} 2 ·2 LOS V = {{{[®c® £4Ð » (H™ • SS₂ ( x² + y²) dA r²r dr do 0 S³ cos" o do $ = 20 cos cos 40 do
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