Variable density in one dimension Find the mass of the following thin bars . 52. A bar on the interval 0 ≤ x ≤ 6 with a density ρ ( x ) = { 1 if 0 ≤ x < 2 2 if 2 ≤ x < 4 4 if 4 ≤ x ≤ 6.
Variable density in one dimension Find the mass of the following thin bars . 52. A bar on the interval 0 ≤ x ≤ 6 with a density ρ ( x ) = { 1 if 0 ≤ x < 2 2 if 2 ≤ x < 4 4 if 4 ≤ x ≤ 6.
Solution Summary: The author explains how the mass of a thin bar or wire is obtained by using the above stated definition.
a. Find the center of mass of a thin plate of constant density cov-ering the region between the curve y = 3/x^3/2 and the x-axis from x = 1 to x = 9.
b. Find the plate’s center of mass if, instead of being constant, the density is d(x) = x. (Use vertical strips).
A tank has a shape of a cone with a radius at the top of 2 m and a height of 5 m. The tank also has a 1 m spout at the top of the tank. The tank is filled with water up to a height of 2 m. Find the work needed to pump all the water out the top of the spout. (Use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)
Consider the center of mass of the following. A lamina occupies the part of the disk x2 + y2 ≤ 36 in the first quadrant. The density at any point is proportional to its distance from the x-axis.
Find the density function. (Use k as the constant of proportionality.)
Find the mass of the entire lamina. (Use k as the constant of proportionality.)
Find the moment of the entire lamina with respect to each axis. (Use k as the constant of proportionality.)
Find the center of mass
Chapter 6 Solutions
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY