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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 6.1.23a
Chapter 6, Problem 6.1.23a

{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. 


a. Determine the position function, for t≥0.

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Recall that velocity is the derivative of the position function with respect to time, so we have \(v(t) = s\'(t)\). Given \(v(t) = 2\pi \cos \pi t\), we want to find \(s(t)\) such that \(s\'(t) = 2\pi \cos \pi t\).
To find \(s(t)\), integrate the velocity function with respect to \(t\): \(s(t) = \int 2\pi \cos \pi t \, dt\).
Use the substitution method for integration: let \(u = \pi t\), so \(du = \pi dt\) or \(dt = \frac{du}{\pi}\). Rewrite the integral in terms of \(u\) and then integrate.
After integrating, substitute back to express \(s(t)\) in terms of \(t\). Remember to include the constant of integration \(C\).
Use the initial condition \(s(0) = 0\) to solve for the constant \(C\) by plugging in \(t=0\) into your expression for \(s(t)\) 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.

Relationship Between Velocity and Position

Velocity is the derivative of the position function with respect to time. To find the position function from velocity, you integrate the velocity function over time, adding a constant of integration determined by initial conditions.
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Derivatives Applied To Velocity

Integration of Trigonometric Functions

Integrating trigonometric functions like cosine involves using standard integral formulas. For example, the integral of cos(ax) is (1/a) sin(ax) + C, where a is a constant. This is essential for finding position from velocity when velocity is given as a cosine function.
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Introduction to Trigonometric Functions

Initial Conditions in Differential Calculus

Initial conditions, such as s(0) = 0, are used to determine the constant of integration after integrating. They ensure the solution matches the physical scenario, providing a unique position function that fits the problem's starting point.
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Related Practice
Textbook Question

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) = 9−t² on [0, 4]; s(0)=−2

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Textbook Question

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The distance traveled by an object moving along a line is the same as the displacement of the object.

Textbook Question

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 


a. Find Q(t), the total amount of oil extracted by the nation after t years.

Textbook Question

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


a. Find the area of region R₁.


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

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.

a. The displacement between t=0 and t=5