EXAMPLE 5 Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder x = 3y2 and the planes x = z, z = 0, and x = 3. SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfac E are the planes z = 0 and z = x, so we describe E as a type 1 region: ✔ Sy≤ 1 ✔3y² ≤ x ≤ 3,0 ≤25x}. Then, if the density is p(x, y, z) = p, the mass is 111₁ pv = [1²₂1²₂² dv Z TLE AIC C = of ² (9 - 9y4) dy P[ E = m = = Y₁. of fol P p dz dx dy 11 X 1x = 3 11:2² x== dy dx dy dy
EXAMPLE 5 Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder x = 3y2 and the planes x = z, z = 0, and x = 3. SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfac E are the planes z = 0 and z = x, so we describe E as a type 1 region: ✔ Sy≤ 1 ✔3y² ≤ x ≤ 3,0 ≤25x}. Then, if the density is p(x, y, z) = p, the mass is 111₁ pv = [1²₂1²₂² dv Z TLE AIC C = of ² (9 - 9y4) dy P[ E = m = = Y₁. of fol P p dz dx dy 11 X 1x = 3 11:2² x== dy dx dy dy
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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Help me find Myz and Mxy
And the center mass
Transcribed Image Text:EXAMPLE 5 Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder
x = 3y² and the planes x = z, z = 0, and x = 3.
SOLUTION The solid E and its projection onto the xy-plane are shown in the figure. The lower and upper surfac
E are the planes z = 0 and z = x, so we describe E as a type 1 region:
✔ ≤y≤ 1
✔3y² ≤ x ≤ 3,0 ≤25x}.
Then, if the density is p(x, y, z) = p, the mass is
11₁₁ Pov = [1²₂6² ₂²
p dv
E =
{(x, y, z) | 1
m =
=
=
=
=
TLE
of fol
P
C
of (9 - 9y4) dy
P[
p dz dx dy
1.
X
1x = 3
11:2²
x==
dy
dx dy
dy
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