GIVEN: a>0, a constant; The finite solid, W, with boundary surfaces ₁, 2 Add extra 9₁: z = a² − (x² + y²) 9₂: z = 0 W is the solid below the paraboloid 2, and above the plane 22. Consider the function f: WC R³ R, f(x, y, z) = z² EVALUATE: Sz² dxdydz. W HINT: Notice that W is a solid of revolution; thus, Try the cylindrical transformation. X 2 pages, as needed. (0, 0, a²) 22₁ Y
GIVEN: a>0, a constant; The finite solid, W, with boundary surfaces ₁, 2 Add extra 9₁: z = a² − (x² + y²) 9₂: z = 0 W is the solid below the paraboloid 2, and above the plane 22. Consider the function f: WC R³ R, f(x, y, z) = z² EVALUATE: Sz² dxdydz. W HINT: Notice that W is a solid of revolution; thus, Try the cylindrical transformation. X 2 pages, as needed. (0, 0, a²) 22₁ Y
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:GIVEN: a>0, a constant; The finite solid, W, with boundary surfaces Q₁, Q2
Add extra
9₁: z = a² - (x² + y²)
92₂: z = 0
W is the solid below the paraboloid 2, and above the plane 22.
Consider the function f: WC R³
R, f(x, y, z) = z²
EVALUATE: Sz² dxdydz.
W
HINT: Notice that W is a solid of revolution; thus,
Try the cylindrical transformation.
X
2
pages, as
needed.
(0, 0, a²)
22₁
Y
![4.5 Consider the finite solid, W, with boundary surfaces
Q₁: z = a²- (x² + y²) for a>0, a constant., and 22: z = 0.
Consider the function f: WC R'
Find √z dxdydz.
1₁:2=a²-(x² + y²), ₁:2=0
√ √ √z dxdydz = SS₁² √z dzdxdy
31279²
= [] [23/1/72] * dudy
=√√ 2 103,5/2² dudy
NOW,
Z=a² - (x² + y²)
0² (x² + y²)=0
(x² + y²)=a²²
Let x = acos (e), y = asin(e)
0≤r≤9, 0≤0≤2T
√w √ Z dxdydz=ff_²3² dxdy
11
20³ 12 rdrde
=
= 20³ 12 [#1 de
2TT
r²
0
2
22π
-
[² [²/²]
de
2
R, f(x, y, z) = √z
2 TT
2T
-1² do = 1/3² [01] "
2π0²
3
Ω
Z
(0,0,a²)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4295bf62-da14-49f1-b3bf-6e678bc68de3%2F369ad9da-902d-4c24-90a0-de0e868e52db%2Fue9b0wp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.5 Consider the finite solid, W, with boundary surfaces
Q₁: z = a²- (x² + y²) for a>0, a constant., and 22: z = 0.
Consider the function f: WC R'
Find √z dxdydz.
1₁:2=a²-(x² + y²), ₁:2=0
√ √ √z dxdydz = SS₁² √z dzdxdy
31279²
= [] [23/1/72] * dudy
=√√ 2 103,5/2² dudy
NOW,
Z=a² - (x² + y²)
0² (x² + y²)=0
(x² + y²)=a²²
Let x = acos (e), y = asin(e)
0≤r≤9, 0≤0≤2T
√w √ Z dxdydz=ff_²3² dxdy
11
20³ 12 rdrde
=
= 20³ 12 [#1 de
2TT
r²
0
2
22π
-
[² [²/²]
de
2
R, f(x, y, z) = √z
2 TT
2T
-1² do = 1/3² [01] "
2π0²
3
Ω
Z
(0,0,a²)
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