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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 6.1.10c
Chapter 6, Problem 6.1.10c

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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c. The position at t=5

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Identify the velocity function v(t) from the graph. Notice that from t=0 to t=3, the velocity is constant at 3 units, and from t=3 to t=5, the velocity decreases linearly from 3 to 0.
Recall that the position function s(t) is the integral of the velocity function v(t) with respect to time t, and the initial position is given as s(0) = 0.
Calculate the position at t=5 by finding the area under the velocity curve from t=0 to t=5. This area represents the displacement.
Break the area under the curve into two parts: a rectangle from t=0 to t=3 with height 3, and a triangle from t=3 to t=5 where velocity decreases from 3 to 0.
Compute the area of the rectangle as base times height: (3 - 0) * 3, and the area of the triangle as (1/2) * base * height: (1/2) * (5 - 3) * 3. Add these two areas to find the total displacement, which equals s(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. To find the position at a given time, you integrate the velocity function over the time interval, starting from the initial position. This means the position is the accumulated area under the velocity-time graph.
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Derivatives Applied To Velocity

Definite Integral as Area Under Curve

The definite integral of a velocity function from time a to b represents the net displacement during that interval. Graphically, this is the area under the velocity curve between t = a and t = b, considering areas above the time-axis as positive and below as negative.
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Definition of the Definite Integral

Piecewise Linear Functions

When velocity is given as a piecewise linear function, the area under the curve can be found by calculating areas of simple geometric shapes like rectangles and triangles. Summing these areas over the intervals gives the total displacement, which helps determine the position.
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Piecewise Functions
Related Practice
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


c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). 

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Textbook Question

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

v(t)={3t if 0≤t<2060 if 20≤t<45240−4t if t≥45v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)

, where is measured in seconds and v has units of m/s. 


c. What is the distance traveled by the automobile in the first 60 s?

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Textbook Question

{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0. 


c. At what times does the mass reach its low point the first three times? 

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Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


c. How far has the cyclist traveled when her velocity is 250 m/min?

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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.


c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

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Textbook Question

Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.

c. How much work is required to stretch the spring 0.3 m from its equilibrium position?

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