[15] (3) GIVEN: constants a > 0 W is the solid cube with edge length, a. 0 ≤ x ≤ a -Jes.,x: 0/10 (x, y, z) 0 ≤ y ≤ a} (x₂) 0 ≤ z ≤ a) Z W = f(x, y, z) = x² + y² + z² FIND: f(x, y, z) dxdydz. X
[15] (3) GIVEN: constants a > 0 W is the solid cube with edge length, a. 0 ≤ x ≤ a -Jes.,x: 0/10 (x, y, z) 0 ≤ y ≤ a} (x₂) 0 ≤ z ≤ a) Z W = f(x, y, z) = x² + y² + z² FIND: f(x, y, z) dxdydz. X
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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Question
for the first image attach please do the calculations similar to the second image attach
please please answer everything correctly I would really appreciate if you would answer
this is not a graded question
![[15] (3)
GIVEN: constants a > 0
W is the solid cube with edge length, a.
0 ≤ x ≤
≤ y ≤ a
0 ≤z≤ a
W has constant density, S, and total mass, M.
Let L be the axis of about which
we want to calculate
the moment of inertia.
GIVEN: L = z-axis.
FIND: The moment of inertia of W w.r.t, L, I₂
(Express I in terms of a and M)
NOTE: V(W) = a³
I₂ = [(x² + y²) S dv
W
=
(x, y, z) 0
=
= 28 √ √ x²dv
W
= 28
=
||
X
s [ x ² dv + S√√ y ² d
S
L = 2-axis
a
a
28 √ ² √ ² √ ªz ² d z dy dz
x
2
Ma²
a
a
2
=
28 (√² x ²4x) ([² 4 ) ( ["dz)
:) (6
25 ( 1 a ²)(a)(a)
5
2011 (10³)
M
a
a³
Z
|
|
|
Note: If W is a solid of revolution about L,
then the computation of I may benefit
from the Cylindrical Transformation.
W
Y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe343d170-4423-4dcd-9c75-f5f0118e0ff9%2Ff4aad326-35da-42c7-abfc-8057e1c15a28%2Fmemw28f_processed.png&w=3840&q=75)
Transcribed Image Text:[15] (3)
GIVEN: constants a > 0
W is the solid cube with edge length, a.
0 ≤ x ≤
≤ y ≤ a
0 ≤z≤ a
W has constant density, S, and total mass, M.
Let L be the axis of about which
we want to calculate
the moment of inertia.
GIVEN: L = z-axis.
FIND: The moment of inertia of W w.r.t, L, I₂
(Express I in terms of a and M)
NOTE: V(W) = a³
I₂ = [(x² + y²) S dv
W
=
(x, y, z) 0
=
= 28 √ √ x²dv
W
= 28
=
||
X
s [ x ² dv + S√√ y ² d
S
L = 2-axis
a
a
28 √ ² √ ² √ ªz ² d z dy dz
x
2
Ma²
a
a
2
=
28 (√² x ²4x) ([² 4 ) ( ["dz)
:) (6
25 ( 1 a ²)(a)(a)
5
2011 (10³)
M
a
a³
Z
|
|
|
Note: If W is a solid of revolution about L,
then the computation of I may benefit
from the Cylindrical Transformation.
W
Y
![[15] (3)
GIVEN: constants a > 0
W is the solid cube with edge length, a.
0 ≤ x ≤
· fer. 3.2
W =
(x, y, z) 0 ≤ y ≤ a
0 ≤
f(x, y, z) = x² + y² + z²
FIND: √ f(x, y, z) dxdydz
VI
X
L = z-axis
Z
W
Y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe343d170-4423-4dcd-9c75-f5f0118e0ff9%2Ff4aad326-35da-42c7-abfc-8057e1c15a28%2F21mjq9g_processed.png&w=3840&q=75)
Transcribed Image Text:[15] (3)
GIVEN: constants a > 0
W is the solid cube with edge length, a.
0 ≤ x ≤
· fer. 3.2
W =
(x, y, z) 0 ≤ y ≤ a
0 ≤
f(x, y, z) = x² + y² + z²
FIND: √ f(x, y, z) dxdydz
VI
X
L = z-axis
Z
W
Y
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