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MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z = 5x2 + 5y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x. SOLUTION The solid lies above the disk D whose boundary circle has equation x2 + y? = 2x or, after completing the square, + y2 = In polar coordinates we have x2 + y? =2 and x = r cos(0), so the boundary circle becomes r² = 2r cos(0), or r= 2 cos(0). Thus the disk D is given by = {(r, 0) -1/2 s 0 S T/2, 0 srs 2 and, by this formula, we have (5x2 + 5y²) dA 1/2 2 cos(@) dr de 2 cos(0) de /2 cos"(0) de T/2 1 + de 7/2 +2 cos(26) + 글(1+ cos(40)| d8 17/2 10

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z = 5x2 + 5y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x. SOLUTION The solid lies above the disk D whose boundary circle has equation x2 + y? = 2x or, after completing the square, + y2 = In polar coordinates we have x2 + y? =2 and x = r cos(0), so the boundary circle becomes r² = 2r cos(0), or r= 2 cos(0). Thus the disk D is given by = {(r, 0) -1/2 s 0 S T/2, 0 srs 2 and, by this formula, we have (5x2 + 5y²) dA 1/2 2 cos(@) dr de 2 cos(0) de /2 cos"(0) de T/2 1 + de 7/2 +2 cos(26) + 글(1+ cos(40)| d8 17/2 10

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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MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z = 5x2 + 5y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x. SOLUTION The solid lies above the disk D whose boundary circle has equation x2 + y? = 2x or, after completing the square, + y2 = In polar coordinates we have x2 + y? =2 and x = r cos(0), so the boundary circle becomes r² = 2r cos(0), or r= 2 cos(0). Thus the disk D is given by = {(r, 0) -1/2 s 0 S T/2, 0 srs 2 and, by this formula, we have (5x + 5y2) dA 1/2 2 cos(@) dr de 12 cos(8) de * T/2 cos (0) de T/2 1 + de 7/2 +2 cos(26) + 글(1+ cos(48)| d8 17/2 10/1
Transcribed Image Text:MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z = 5x2 + 5y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x. SOLUTION The solid lies above the disk D whose boundary circle has equation x2 + y? = 2x or, after completing the square, + y2 = In polar coordinates we have x2 + y? =2 and x = r cos(0), so the boundary circle becomes r² = 2r cos(0), or r= 2 cos(0). Thus the disk D is given by = {(r, 0) -1/2 s 0 S T/2, 0 srs 2 and, by this formula, we have (5x + 5y2) dA 1/2 2 cos(@) dr de 12 cos(8) de * T/2 cos (0) de T/2 1 + de 7/2 +2 cos(26) + 글(1+ cos(48)| d8 17/2 10/1
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