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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 6.1.19a
Chapter 6, Problem 6.1.19a

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) = 6−2t on [0, 5]; s(0)=0

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1
Identify the given velocity function: \(v(t) = 6 - 2t\) and the initial position \(s(0) = 0\).
Recall that the position function \(s(t)\) is the antiderivative (indefinite integral) of the velocity function \(v(t)\), so write \(s(t) = \int v(t) \, dt + C\).
Integrate the velocity function: compute \(\int (6 - 2t) \, dt\) by integrating each term separately.
After integration, express \(s(t)\) as \(s(t) = 6t - t^2 + C\), where \(C\) is the constant of integration.
Use the initial condition \(s(0) = 0\) to solve for \(C\) by substituting \(t=0\) into \(s(t)\) and setting it equal to zero.

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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 from the initial point.
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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. Finding the antiderivative of velocity gives the position function up to a constant, which 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 from velocity, an unknown constant appears. The initial position value (e.g., s(0) = 0) is used 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

55–58. Marginal cost Consider the following marginal cost functions.


a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.


C′(x)=200−0.05x

Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

Textbook Question

Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).

a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

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Textbook Question

A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.

a. How much work is required to compress the spring and lift the mass 0.5 m?

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Textbook Question

Region R is revolved about the line x=4 to form a solid of revolution.


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

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

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