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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.1.65c
Capitolo 6, Problema 6.1.65c

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). 

Guida verificata passo dopo passo
1
Understand that the distance each rider travels is the integral of their velocity function over the time interval from 0 to 2 hours. This is because distance is the area under the velocity-time graph.
For Theo, whose velocity is constant at \(v_T(t) = 10\) mi/hr, calculate the distance by integrating the constant velocity over 2 hours: \(\int_0^2 10 \, dt\).
For Sasha, whose velocity changes, split the integral into two parts: from 0 to 1 hour where \(v_S(t) = 15t\), and from 1 to 2 hours where \(v_S(t) = 15\). So, calculate \(\int_0^1 15t \, dt + \int_1^2 15 \, dt\).
Evaluate both integrals (without computing the final numerical values here) to find the total distance each rider covers in 2 hours.
Compare the two distances to determine who rides farther. Geometrically, this corresponds to comparing the areas under each velocity curve on the graph from 0 to 2 hours.

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Velocity as a Function of Time

Velocity describes how fast an object moves and in which direction, often expressed as a function of time. Understanding velocity functions allows us to analyze how speed changes over time, such as Theo's constant velocity and Sasha's piecewise velocity that increases then remains constant.
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Using The Velocity Function

Distance Traveled as the Integral of Velocity

The total distance traveled over a time interval is found by integrating the velocity function over that interval. This means calculating the area under the velocity-time graph, which represents the accumulation of movement over time.
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Using The Velocity Function

Interpreting Graphs of Piecewise Functions

Piecewise functions have different expressions over different intervals. Interpreting their graphs involves understanding how the function changes shape, such as Sasha's velocity increasing linearly then becoming constant, which affects the area under the curve and thus the total distance.
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Piecewise Functions
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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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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.

c. The position at t=5

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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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Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours. 


c. What is the hiker’s position at t=3?

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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.


c. Find the distance traveled over the given interval.


v(t) = 3t²−6t on [0, 3]

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Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.


c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

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