4. A metal piece with variable density has the shape of a parallelepiped withsides equal to 3, 4 and 5 cm, respectively. What is the mass of the piece if thedensity function (in dg/cm3) is given by:p(x, y, z) = 2x + xy² + z + 1; 0 ≤ x ≤ 3; 0 ≤ y ≤ 4; 0 ≤ z ≤ 5?

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Answered: 4. A metal piece with variable density…

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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2. Suppose a solid mass is in the shape of the unit ball in R°, that is, the region with 2² + y? + 2?< 1. Suppose its mass density is f(x, y, z) = 1- Vr² + y? + z?. Find its total mass.
A cylindrical tank having a radius of 0.5meters and a height of 2 meters is half-full ofkerosene, which has a density of 810 kg per?3. Find the work needed to pump thekerosene over the top edge of the tank.
Consider the plate bounded by y² = 8x and x = 2 with density 8 = 2 - x. 1. Sketch a graph of the plate. Shade in the region and number the axes so that the important points on the plate can be read from the graph. 2. Express the mass of the plate as a double integral and evaluate to find the mass. 3. Express the x-coordinate of the center of mass of the plate as a double integral and evaluate to find the value of . As a side note, by the symmetry of the plate and density, the y-coordinate of the center of mass y = 0. It might be good practice to confirm this though.
2. Suppose you have a conical tank of height 4m and radius 1m filled with a liquid of density p (shown below). R = 1 H = 4
Find the mass of the lamina in the first quadrant bounded by y = x3 and y = x. The density at any point %3D (x, y) is directly proportional to twice the distance from the x-axis. Exact answers only, no calculator answers.
2. A lamina occupies the region inside the triangle (shown below). Suppose that the density at point (z, y) on the lamina is given by p(x, y) = x²+y (grams per square centimeter). Find the total mass of the lamina. %3D 3 (2, 3) 1 2

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4. A metal piece with variable density has the shape of a parallelepiped with sides equal to 3, 4 and 5 cm, respectively. What is the mass of the piece if the density function (in dg/cm3) is given by: p(x, y, z) = 2x + xy +z+1; 0≤x≤ 3; 0 ≤ y ≤ 4; 0 ≤ z ≤ 5?