3 You do not need to show your work. (a) Let D be the region inside the unit circle centered at the origin, let R be the right half of D and let B be the bottom part of D. Decide (without calculation) whether the integrals below are positive, negative, or zero. (i) Spex+y dA is (A) positive (B) negative (C) zero. (ii) JB x cos(y) dA is (A) positive (B) negative (C) zero. (iii) R(x³-x)99 dA is (A) positive (B) negative (C) zero. √1-x2 (b) The integral LIEL -(x²+y²+z²) dz dy dx describes the mass of √1-22 e (i) a cylinder that gets heavier towards the outside. (ii) a sphere that gets heavier towards the outside. (iii) a cylinder that gets lighter towards the outside. (iv) a sphere that gets lighter towards the outside. (c) Consider the sector of the unit circle R = {(r, 0) : 0 ≤ r ≤ 1, 0 ≤ 0 ≤ 0₁}, where 000 < 0₁ ≤ 2π are fixed angles. Assume that, at each point, the density of R is twice the distance from that point to the origin. In each of the following cases, set up an iterated integral, and evaluate it. (i) The x-coordinate of the center of mass of R is x = (ii) The y-coordinate of the center of mass of R is y =

3 You do not need to show your work. (a) Let D be the region inside the unit circle centered at the origin, let R be the right half of D and let B be the bottom part of D. Decide (without calculation) whether the integrals below are positive, negative, or zero. (i) Spex+y dA is (A) positive (B) negative (C) zero. (ii) JB x cos(y) dA is (A) positive (B) negative (C) zero. (iii) R(x³-x)99 dA is (A) positive (B) negative (C) zero. √1-x2 (b) The integral LIEL -(x²+y²+z²) dz dy dx describes the mass of √1-22 e (i) a cylinder that gets heavier towards the outside. (ii) a sphere that gets heavier towards the outside. (iii) a cylinder that gets lighter towards the outside. (iv) a sphere that gets lighter towards the outside. (c) Consider the sector of the unit circle R = {(r, 0) : 0 ≤ r ≤ 1, 0 ≤ 0 ≤ 0₁}, where 000 < 0₁ ≤ 2π are fixed angles. Assume that, at each point, the density of R is twice the distance from that point to the origin. In each of the following cases, set up an iterated integral, and evaluate it. (i) The x-coordinate of the center of mass of R is x = (ii) The y-coordinate of the center of mass of R is 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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