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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 6.1.26c
Chapter 6, Problem 6.1.26c

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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Step 1: Understand the problem. The hiker's position at time t=3 can be found by integrating the velocity function v(t) = 3 sin²(πt/2) over the interval [0, 3]. This is because position is the integral of velocity with respect to time.
Step 2: Set up the integral. The position function s(t) is given by s(t) = ∫ v(t) dt. Since s(0) = 0, the position at t=3 is s(3) = ∫[0 to 3] 3 sin²(πt/2) dt.
Step 3: Simplify the integrand using a trigonometric identity. Recall that sin²(x) can be rewritten as (1 - cos(2x))/2. Substituting this into the integral, we get s(3) = ∫[0 to 3] 3 * (1 - cos(πt))/2 dt.
Step 4: Break the integral into two parts. The integral becomes s(3) = (3/2) ∫[0 to 3] 1 dt - (3/2) ∫[0 to 3] cos(πt) dt. Evaluate each part separately.
Step 5: Evaluate the integrals. For the first term, ∫[0 to 3] 1 dt is straightforward and equals t evaluated from 0 to 3. For the second term, ∫[0 to 3] cos(πt) dt requires substitution u = πt, du = π dt, and then solving the integral. Combine the results to find s(3).

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Key Concepts

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

Velocity and Position Functions

In calculus, velocity is the rate of change of position with respect to time. The position function, s(t), can be derived from the velocity function, v(t), by integrating it over time. In this case, the hiker's position at any time t can be found by integrating the given velocity function v(t) = 3 sin²(πt/2) from 0 to t.
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Integration

Integration is a fundamental concept in calculus that allows us to find the accumulated value of a function over an interval. To find the hiker's position at t=3, we need to compute the definite integral of the velocity function from 0 to 3. This process essentially sums up all the small changes in position over the specified time period.
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Integration by Parts for Definite Integrals

Definite Integral

A definite integral calculates the net area under a curve between two points on the x-axis. In this context, it represents the total distance traveled by the hiker from time t=0 to t=3. The result of the definite integral will give us the hiker's position at t=3, which is crucial for answering the question.
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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

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)

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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

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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Textbook Question

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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Textbook Question

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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