Skip to main content
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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
06:21
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.
Recommended video:
05:43
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.
Recommended video:
05:36
Piecewise Functions
Related Practice
Textbook Question

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

1
views
Textbook Question

52–54. Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 


The window is circular, with a radius of 0.5 m, tangent to the bottom of the pool.

Textbook Question

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


The region in the first quadrant bounded by y=x^2/3 and y=4

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


x=2−secy,x=2,y=π/3, and y=0; about x=2

1
views
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