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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.RE.72
Chapter 6, Problem 6.RE.72

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Separation of Variables

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.
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Definition of the Definite Integral

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.
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Piecewise Functions
Related Practice
Textbook Question

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d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.


v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If water flows into a tank at a constant rate (for example, 6 gal/min), the volume of water in the tank increases according to a linear function of time.

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Textbook Question

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

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Textbook Question

Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by

R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)

2 if t> 0 (cruising)

a. Find a function R that gives the total fuel consumed, for 0≤t≤8.

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Textbook Question

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.


a. Using calculus

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