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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 \(-25 e^{-8} - (-25 e^{0})\).

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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.
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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.
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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.
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Exponential Growth & Decay
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.


b. What is the height of a cylindrical shell at a point x in [0, 4]?

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Textbook Question

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

b. What is the displacement of the object over the interval [2, 6]? 

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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

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Textbook Question

Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.

b. How much work is done in compressing the spring 0.5 m from its equilibrium position?

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Textbook Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?

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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?

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