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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 6.1.18a
Chapter 6, Problem 6.1.18a

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = −t³+3t²−2t on [0, 3]; s(0)=4

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Identify the given velocity function: \(v(t) = -t^{3} + 3t^{2} - 2t\) and the initial position \(s(0) = 4\).
Recall that the position function \(s(t)\) is the antiderivative (indefinite integral) of the velocity function \(v(t)\), plus a constant of integration \(C\): \(s(t) = \int v(t) \, dt + C\).
Integrate the velocity function term-by-term: \(\int (-t^{3} + 3t^{2} - 2t) \, dt = \int -t^{3} \, dt + \int 3t^{2} \, dt - \int 2t \, dt\).
Calculate each integral separately: \(\int -t^{3} \, dt = -\frac{t^{4}}{4}\), \(\int 3t^{2} \, dt = t^{3}\), \(\int -2t \, dt = -t^{2}\).
Combine the results and add the constant \(C\): \(s(t) = -\frac{t^{4}}{4} + t^{3} - t^{2} + C\). Use the initial condition \(s(0) = 4\) to solve for \(C\) by substituting \(t=0\) and \(s(0)=4\).

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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 can be found by integrating the velocity function over time, which accumulates the changes in position. Understanding this relationship is essential to move from velocity to position.
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Derivatives Applied To Velocity

Antiderivative (Indefinite Integral)

The antiderivative of a function is another function whose derivative is the original function. To find the position from velocity, you compute the antiderivative of the velocity function, adding a constant of integration that can be determined using initial conditions.
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Introduction to Indefinite Integrals

Initial Conditions and Constants of Integration

When integrating to find position, an unknown constant appears. The initial position value, such as s(0) = 4, allows you to solve for this constant, ensuring the position function accurately reflects the object's starting point.
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Initial Value Problems
Related Practice
Textbook Question

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?

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Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

Textbook Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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Textbook Question

"Determine whether the following statements are true and give an explanation or counterexample.


a. A pyramid is a solid of revolution. "

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).