A. Set up the Iterated Triple Integrals in both (i) rectangular coordinates, and (ii) cylindrical coordinates, that represent the mass m and the first-order moments Mry, Mrz, and Myz of the solid region Q with density p(x,y, z) = 2+ 6z, above the ry-plane, and below the cone z = 3 – Vx2 + y². Include limits of integration. B. Evaluate the four integrals found in part A using cylindrical coordinates. C. Determine the coordinates (î, §, z) of the center of mass of Q.
A. Set up the Iterated Triple Integrals in both (i) rectangular coordinates, and (ii) cylindrical coordinates, that represent the mass m and the first-order moments Mry, Mrz, and Myz of the solid region Q with density p(x,y, z) = 2+ 6z, above the ry-plane, and below the cone z = 3 – Vx2 + y². Include limits of integration. B. Evaluate the four integrals found in part A using cylindrical coordinates. C. Determine the coordinates (î, §, z) of the center of mass of Q.
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
parts a - c in pic
Transcribed Image Text:A. Set up the Iterated Triple Integrals in both
(i) rectangular coordinates, and
(ii) cylindrical coordinates,
that represent the mass m and the first-order moments Mru, Mrz, and Muz of the solid region
Q with density p(x, y, z) = 2 + 6z, above the ry-plane, and below the cone
z = 3 – Vx2 + y2.
Include limits of integration.
B. Evaluate the four integrals found in part A using cylindrical coordinates.
C. Determine the coordinates (, j, z) of the center of mass of Q.
Hint: The mass m and the first-order moments My, Mrz, and Myz of a solid region Q with
density p(x, y, z) are given by
I| P(x, y, z) dx dy dz,
m =
and
I| z p(x, y, z) da dy dz, Maz = ||/ y p(x, y, z) dæ dy dz, Myzi
Mry
= /|| x p(x, y, z) dx dy dz.
Q
Q
The coordinates (, j, z) of the center of mass are defined as
Mry
Myz
Maz
m
m
m
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