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 6.1.16b
Chapter 6, Problem 6.1.16b

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]

Verified step by step guidance
1
Identify the displacement as the definite integral of the velocity function over the given time interval. Displacement is given by \(\int_{a}^{b} v(t) \, dt\), where \(a=0\) and \(b=4\) in this problem.
Write down the integral to find displacement: \(\int_{0}^{4} 50 e^{-2t} \, dt\).
Recall the integral formula for an exponential function: \(\int e^{kt} \, dt = \frac{1}{k} e^{kt} + C\). Here, \(k = -2\).
Apply the integral formula to \(50 e^{-2t}\): the antiderivative is \(50 \times \frac{1}{-2} e^{-2t} = -25 e^{-2t}\).
Evaluate the definite integral by substituting the limits: calculate \([-25 e^{-2t}]\) from \(t=0\) to \(t=4\), which means computing \(-25 e^{-8} - (-25 e^{0})\).

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.

Displacement and Velocity Relationship

Displacement represents the change in position of an object and is found by integrating the velocity function over a given time interval. Since velocity is the rate of change of position, integrating velocity with respect to time gives the net displacement.
Recommended video:
06:29
Derivatives Applied To Velocity

Definite Integral

A definite integral calculates the accumulated quantity, such as displacement, over a specific interval. For velocity functions, the definite integral from time a to b gives the total displacement between those times.
Recommended video:
05:43
Definition of the Definite Integral

Exponential Decay Function

The velocity function v(t) = 50e^(-2t) is an exponential decay, meaning velocity decreases rapidly over time. Understanding how to integrate exponential functions is essential to find displacement accurately.
Recommended video:
09:29
Exponential Growth & Decay
Related Practice
Textbook Question

Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

23
views
Textbook Question

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

138
views
Textbook Question

21–30. {Use of Tech} Arc length by calculator


b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

49
views
Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

65
views
Textbook Question

A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.

a. How much work is required to compress the spring and lift the mass 0.5 m?

62
views
Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.

51
views