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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 6.1.9a
Chapter 6, Problem 6.1.9a

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.
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a. The displacement between t=0 and t=5

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Identify that displacement over a time interval is the integral of the velocity function over that interval. In this case, displacement from \(t=0\) to \(t=5\) is \(\int_0^5 v(t) \, dt\).
Observe the velocity graph and break the interval from \(t=0\) to \(t=5\) into segments where the velocity function is piecewise linear or constant. Here, the segments are from \(t=0\) to \(t=2\), \(t=2\) to \(t=3\), and \(t=3\) to \(t=5\).
Calculate the area under the velocity curve for each segment. Positive areas correspond to motion in the positive direction, and negative areas correspond to motion in the negative direction. Use geometric shapes (triangles and rectangles) to find these areas.
Sum the areas from all segments, taking care to add positive and negative areas accordingly. This sum represents the net displacement from \(t=0\) to \(t=5\).
Remember that since the initial position \(s(0) = 0\), the displacement calculated is the change in position \(s(5) - s(0)\).

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Key Concepts

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

Velocity and Displacement

Velocity represents the rate of change of position with respect to time and can be positive or negative depending on direction. Displacement is the net change in position over a time interval and is found by integrating the velocity function over that interval.
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Using The Velocity Function

Area Under the Velocity-Time Graph

The displacement over a time interval is given by the area under the velocity-time curve between those times. Positive areas correspond to motion in the positive direction, while negative areas correspond to motion in the opposite direction.
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Estimating the Area Under a Curve Using Left Endpoints

Piecewise Linear Functions and Integration

When the velocity graph is piecewise linear, displacement can be calculated by finding the areas of geometric shapes (triangles, rectangles) under the curve and summing them, considering their signs. This approach simplifies integration for such graphs.
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Piecewise Functions
Related Practice
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1
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Textbook Question

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