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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 6.1.8e
Chapter 6, Problem 6.1.8e

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.


e. Describe the position of the object relative to its initial position after 8 seconds.

Verified step by step guidance
1
Identify that the position of the object relative to its initial position after a certain time is given by the displacement, which is the integral of the velocity function over that time interval.
Recall that displacement over the interval from \(t = 0\) to \(t = 8\) seconds is calculated by integrating the velocity function \(v(t)\) with respect to time: \(\displaystyle \text{Displacement} = \int_0^8 v(t) \, dt\).
Use the given areas of the regions bounded by the velocity curve and the \(t\)-axis to evaluate this integral. Remember that areas above the \(t\)-axis correspond to positive velocity (positive contribution to displacement), and areas below correspond to negative velocity (negative contribution to displacement).
Sum the signed areas (positive areas minus negative areas) to find the net displacement after 8 seconds, which tells you how far and in which direction the object is from its initial position.
Interpret the result: if the net displacement is positive, the object is to the right (or forward) of its initial position; if negative, it is to the left (or backward); if zero, it has returned to its starting point.

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Key Concepts

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

Velocity and its Relation to Position

Velocity is the rate of change of position with respect to time. The position function can be found by integrating the velocity function over time, which accumulates the net displacement from the initial position.
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Intro To Related Rates

Displacement vs. Distance

Displacement is the net change in position, considering direction, while distance is the total length traveled regardless of direction. Displacement can be positive, negative, or zero, whereas distance is always non-negative.
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Using The Acceleration Function Example 1

Area Under the Velocity Curve

The area between the velocity curve and the time-axis represents displacement over a time interval. Positive areas indicate movement in the positive direction, and negative areas indicate movement in the opposite direction; summing these areas gives the net displacement.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
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.


e. How far has the racer traveled when it reaches a speed of 178 ft/s?

Textbook Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

Textbook Question

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.

d. A piecewise function for s(t)

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.

d. What is the displacement of the object over the interval [0, 8]? 

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.


d. How long does it take the racer to travel 300 ft?