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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 6.7.18
Chapter 6, Problem 6.7.18

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.


ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3

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Identify the density function \( \rho(x) \) given as a piecewise function: \( \rho(x) = 1 \) for \( 0 \leq x \leq 2 \) and \( \rho(x) = 2 \) for \( 2 < x \leq 3 \).
Recall that the mass \( m \) of a thin bar along the x-axis with density function \( \rho(x) \) is found by integrating the density over the length of the bar: \( m = \int_a^b \rho(x) \, dx \).
Since the density function changes at \( x=2 \), split the integral into two parts: \( m = \int_0^2 1 \, dx + \int_2^3 2 \, dx \).
Set up each integral separately: the first integral calculates the mass from 0 to 2 with density 1, and the second integral calculates the mass from 2 to 3 with density 2.
Evaluate each integral by integrating the constant densities over their respective intervals, then sum the results to find the total mass of the bar.

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Key Concepts

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

Density Function

The density function ρ(x) describes how mass is distributed along the length of the bar. It can vary with position x, indicating that different segments of the bar may have different mass per unit length. Understanding this function is essential to calculate the total mass accurately.
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Properties of Functions

Definite Integral for Mass

The total mass of a one-dimensional object with variable density is found by integrating the density function over the object's length. This involves computing the definite integral of ρ(x) with respect to x over the given intervals, summing contributions from each segment.
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Definition of the Definite Integral

Piecewise Functions

When the density function is defined differently over separate intervals, it is called a piecewise function. To find the total mass, you must integrate each piece over its respective interval and then add the results, ensuring correct handling of the function's discontinuities.
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Piecewise Functions
Related Practice
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The window is circular, with a radius of 0.5 m, tangent to the bottom of the pool.

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Find the area of the region described in the following exercises.


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Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.


x =2

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Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.

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Textbook Question

Let f(x) = {x   if 0≤x≤2

      2x−2  if 2<x≤5

      −2x+18 if 5<x≤6. 


Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.

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