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 6.1.12c
Chapter 6, Problem 6.1.12c

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


c. Find the distance traveled by the object on the interval 0≤t≤6.

Verified step by step guidance
1
Understand the difference between displacement and distance traveled: displacement is the net change in position, while distance traveled is the total length of the path covered, regardless of direction.
Identify the velocity function given: \(v(t) = 6 - 2t\) for \(0 \leq t \leq 6\). To find distance traveled, we need to consider when the velocity changes sign because the object might change direction.
Find the time when the velocity is zero by solving \(6 - 2t = 0\). This gives the critical point where the object changes direction.
Split the interval \([0,6]\) into subintervals based on the critical point found. On each subinterval, the velocity has a consistent sign, so integrate the absolute value of velocity over each subinterval to find the distance traveled in that segment.
Calculate the total distance traveled by summing the absolute values of the integrals of velocity over each subinterval: \(\text{Distance} = \int_0^{t_0} |v(t)| \, dt + \int_{t_0}^6 |v(t)| \, dt\), where \(t_0\) is the time when velocity is zero.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Velocity and Displacement

Velocity is the rate of change of position with respect to time and can be positive or negative, indicating direction. Displacement is the net change in position over a time interval and is found by integrating velocity. Understanding the difference between velocity and displacement is crucial for analyzing motion.
Recommended video:
10:17
Using The Velocity Function

Distance Traveled vs. Displacement

Distance traveled is the total length of the path covered, regardless of direction, while displacement is the straight-line change in position. To find distance traveled from velocity, one must integrate the absolute value of velocity over the time interval, accounting for any changes in direction.
Recommended video:
08:56
Using The Acceleration Function Example 1

Definite Integrals and Absolute Value

Definite integrals calculate the accumulation of quantities over an interval. When velocity changes sign, the integral of velocity gives displacement, but to find total distance, the integral of the absolute value of velocity is needed. This requires identifying intervals where velocity is positive or negative and integrating accordingly.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

c. Write an integral for the volume of the solid using the shell method.

Textbook Question

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 


c. At what times does the mass reach its low point the first three times? 

Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


c. How far has the cyclist traveled when her velocity is 250 m/min?

1
views
Textbook Question

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

c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

Textbook Question

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


Arc length may be negative if f(x) < 0 on part of the interval in question.

Textbook Question

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.


c. At this rate, how long will it take the racer to travel 1/4 mi?

1
views