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 6.1.12b
Chapter 6, Problem 6.1.12b

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.


b. Find the displacement of the object on the interval 0≤t≤6.

Verified step by step guidance
1
Recall that displacement over a time interval is the net change in position, which can be found by integrating the velocity function over that interval.
Set up the definite integral for displacement using the given velocity function \(v(t) = 6 - 2t\) over the interval \(0 \leq t \leq 6\): \[\int_0^6 (6 - 2t) \, dt\]
Integrate the function \(6 - 2t\) with respect to \(t\). The integral of \(6\) is \$6t\(, and the integral of \)-2t\( is \)-t^2$.
Evaluate the antiderivative at the upper and lower limits of the interval, \(t=6\) and \(t=0\), and subtract to find the displacement: \[\left[6t - t^2\right]_0^6 = (6 \times 6 - 6^2) - (6 \times 0 - 0^2)\]
Simplify the expression to find the net displacement of the object over the time interval.

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 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 the velocity function over that interval.
Recommended video:
10:17
Using The Velocity Function

Definite Integral as Net Change

The definite integral of a velocity function from time a to b gives the net change in position, or displacement, during that time. It sums the signed areas under the velocity curve, accounting for direction.
Recommended video:
05:43
Definition of the Definite Integral

Evaluating Definite Integrals

To find displacement, compute the definite integral of v(t) from 0 to 6 by finding an antiderivative and applying the Fundamental Theorem of Calculus. This involves substituting the limits into the antiderivative and subtracting.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

40–43. Population growth


A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.


a. What is the population after 20 days? After 40 days?

Textbook Question

40–43. Population growth


Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.


b. Find the population P(t), for 0≤t≤200.

Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m from its equilibrium position?

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