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 6.7.20
Chapter 6, Problem 6.7.20

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


ρ(x) = {x² if 0≤x≤1 {x(2-x) if 1<x≤2

Verified step by step guidance
1
Identify the density function \( \rho(x) \) given piecewise as \( \rho(x) = x^2 \) for \( 0 \leq x \leq 1 \) and \( \rho(x) = x(2 - x) \) for \( 1 < x \leq 2 \).
Recall that the mass \( M \) of a thin bar along the x-axis from \( a \) to \( b \) with density function \( \rho(x) \) is given by the integral \( M = \int_a^b \rho(x) \, dx \).
Since the density function is piecewise, split the integral into two parts: \( M = \int_0^1 x^2 \, dx + \int_1^2 x(2 - x) \, dx \).
Set up each integral separately: first, \( \int_0^1 x^2 \, dx \), and second, \( \int_1^2 (2x - x^2) \, dx \) after expanding the second density function.
Evaluate each integral individually and then add 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 and Mass

The density function ρ(x) describes how mass is distributed along a one-dimensional object. To find the total mass, you integrate the density over the length of the object, summing the infinitesimal masses at each point.
Recommended video:
06:21
Properties of Functions

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. When integrating, you must split the integral at the points where the function changes to correctly calculate quantities like mass.
Recommended video:
05:36
Piecewise Functions

Definite Integration

Definite integration calculates the accumulation of quantities, such as mass, over an interval. For this problem, integrating the density function over the specified intervals yields the total mass of the bar.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

Determine the area of the shaded region in the following figures.

112
views
Textbook Question

Work from force How much work is required to move an object from x=0 to x=3 (measured in meters) in the presence of a force (in N) given by F(x)=2x acting along the x-axis?

48
views
Textbook Question

Drinking juice A glass has circular cross sections that taper (linearly) from a radius of 5 cm at the top of the glass to a radius of 4 cm at the bottom. The glass is 15 cm high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is 5 cm above the top of the glass? Assume the density of orange juice equals the density of water.

62
views
Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

90
views
Textbook Question

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]

72
views
Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.

79
views