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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 6.1.10b
Chapter 6, Problem 6.1.10b

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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b. The distance traveled between t=0 and t=5

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Identify the velocity function from the graph. From t = 0 to t = 3, the velocity is constant at 3 units per time. From t = 3 to t = 5, the velocity decreases linearly from 3 to 0.
Recall that the distance traveled is the total length of the path, which is the integral of the absolute value of velocity over the time interval. Since velocity is positive here, distance traveled equals the integral of velocity from t = 0 to t = 5.
Break the integral into two parts corresponding to the two segments of the velocity graph: from 0 to 3 and from 3 to 5. So, calculate \( \int_0^3 3 \, dt \) and \( \int_3^5 v(t) \, dt \), where \( v(t) \) is the linear function decreasing from 3 to 0.
For the second part, find the equation of the line for velocity between t = 3 and t = 5. Use the two points (3, 3) and (5, 0) to find the slope and write \( v(t) = m t + b \).
Calculate each integral separately and then add the results to find the total distance traveled between t = 0 and t = 5.

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Key Concepts

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

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. The position function s(t) can be found by integrating the velocity function v(t). Understanding this relationship allows us to determine how far an object has moved over a time interval.
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Derivatives Applied To Velocity

Distance Traveled vs. Displacement

Distance traveled is the total length of the path taken, regardless of direction, while displacement is the net change in position. When velocity changes sign, distance traveled is found by integrating the absolute value of velocity, ensuring all movement contributes positively.
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Using The Acceleration Function Example 1

Area Under the Velocity Curve

The area under the velocity-time graph between two times represents displacement. For distance traveled, the total area between the curve and the time axis is considered, treating areas below the axis as positive. This geometric interpretation helps calculate distance from velocity graphs.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
Textbook Question

Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.


b. How far does the car travel in the time it takes to come to rest?

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Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


b. What is the SAV ratio of a ball with radius a? 

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Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?

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Textbook Question

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=tan x , for 0≤x≤π/4; about the x-axis 

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Textbook Question

40–43. Population growth


When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.


b. Find the population P(t) at any time t≥0.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

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