70–72. Variable density in one dimension Find the mass of the following thin bars.

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

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Identify the intervals and corresponding density functions given for the bar: \(0 \leq x < 2\) with density \(\rho(x) = 1\), \(2 \leq x < 4\) with density \(\rho(x) = 2\), and \(4 \leq x \leq 6\) with density \(\rho(x) = 4\).

Recall that the mass of a thin bar with variable density \(\rho(x)\) over an interval \([a,b]\) is found by integrating the density function over that interval: \[ m = \int_a^b \rho(x) \, dx \].

Since the density is piecewise constant, split the integral into three parts corresponding to the intervals: \[ m = \int_0^2 1 \, dx + \int_2^4 2 \, dx + \int_4^6 4 \, dx \].

Evaluate each integral separately by integrating the constant densities over their respective intervals: \[ \int_0^2 1 \, dx, \quad \int_2^4 2 \, dx, \quad \int_4^6 4 \, dx \].

Sum the results of the three integrals to find the total mass of the bar: \[ m = \text{(result of first integral)} + \text{(result of second integral)} + \text{(result of third integral)} \].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Variable Density Function

A variable density function ρ(x) describes how mass per unit length changes along the bar. In this problem, the density is piecewise constant, meaning it takes different constant values on different intervals. Understanding this helps set up the correct integral for each segment.

Definite Integral for Mass Calculation

The mass of a thin bar with variable density is found by integrating the density function over the length of the bar. Specifically, mass = ∫ ρ(x) dx over the given interval. For piecewise functions, the integral is split into parts corresponding to each density segment.

Piecewise Integration

When the density function is defined in pieces over different intervals, the total mass is the sum of integrals over each interval. This requires evaluating separate integrals for each density value and then adding the results to find the total mass.

70–72. Variable density in one dimension Find the mass of the following thin bars.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