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 6.1.10b
Chapter 6, Problem 6.1.10b

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
img
b. The distance traveled between t=0 and t=5

Verified step by step guidance
1
Identify the velocity function from the graph. From t = 0 to t = 3, the velocity is constant at 3 units per time. From t = 3 to t = 5, the velocity decreases linearly from 3 to 0.
Recall that the distance traveled is the total length of the path, which is the integral of the absolute value of velocity over the time interval. Since velocity is positive here, distance traveled equals the integral of velocity from t = 0 to t = 5.
Break the integral into two parts corresponding to the two segments of the velocity graph: from 0 to 3 and from 3 to 5. So, calculate \( \int_0^3 3 \, dt \) and \( \int_3^5 v(t) \, dt \), where \( v(t) \) is the linear function decreasing from 3 to 0.
For the second part, find the equation of the line for velocity between t = 3 and t = 5. Use the two points (3, 3) and (5, 0) to find the slope and write \( v(t) = m t + b \).
Calculate each integral separately and then add the results to find the total distance traveled between t = 0 and t = 5.

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.

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. The position function s(t) can be found by integrating the velocity function v(t). Understanding this relationship allows us to determine how far an object has moved over a time interval.
Recommended video:
06:29
Derivatives Applied To Velocity

Distance Traveled vs. Displacement

Distance traveled is the total length of the path taken, regardless of direction, while displacement is the net change in position. When velocity changes sign, distance traveled is found by integrating the absolute value of velocity, ensuring all movement contributes positively.
Recommended video:
08:56
Using The Acceleration Function Example 1

Area Under the Velocity Curve

The area under the velocity-time graph between two times represents displacement. For distance traveled, the total area between the curve and the time axis is considered, treating areas below the axis as positive. This geometric interpretation helps calculate distance from velocity graphs.
Recommended video:
05:59
Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.


b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.


C′(x)=200−0.05x

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

1
views
Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=tan x , for 0≤x≤π/4; about the x-axis 

1
views
Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

1
views
Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 50e^−2t on [0, 4]

Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=cos x, for 0≤x≤π/2; about the x-axis