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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Concept explainers

Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.

Question

parts a - c in pic

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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