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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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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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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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Related Practice
Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

a. On what intervals is the object moving in the positive direction?

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.

Textbook Question

Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.


f. Graph ymax as a function of v0. What is the maximum height when v0=500 m/s,1500 m/s, and 5 km/s?

Textbook Question

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

b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.

Textbook Question

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

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Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = cos 2x, for 0 ≤ x ≤ π