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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 6.1.35
Chapter 6, Problem 6.1.35

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).


a(t) = cos2t; v(0) = 5; s(0) = 7

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1
Identify the given acceleration function: \(a(t) = \cos^2(t)\), the initial velocity \(v(0) = 5\), and the initial position \(s(0) = 7\).
Recall that velocity is the integral of acceleration with respect to time: \(v(t) = v(0) + \int_0^t a(u) \, du\).
Set up the integral for velocity: \(v(t) = 5 + \int_0^t \cos^2(u) \, du\). To solve this integral, use the trigonometric identity \(\cos^2(u) = \frac{1 + \cos(2u)}{2}\) to simplify the integrand.
Once you find \(v(t)\), find the position function by integrating velocity: \(s(t) = s(0) + \int_0^t v(u) \, du\).
Set up the integral for position: \(s(t) = 7 + \int_0^t v(u) \, du\). After computing this integral, you will have the position function \(s(t)\).

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Key Concepts

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

Acceleration, Velocity, and Position Relationship

Acceleration is the derivative of velocity with respect to time, and velocity is the derivative of position. To find velocity from acceleration, integrate the acceleration function, and to find position from velocity, integrate the velocity function. Initial conditions help determine the constants of integration.
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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that integration can be reversed by differentiation. It allows us to find a function given its derivative by integrating, and use initial values to solve for constants, which is essential for finding velocity and position from acceleration.
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Fundamental Theorem of Calculus Part 1

Initial Conditions and Constants of Integration

When integrating acceleration to find velocity, and velocity to find position, each integration introduces an unknown constant. Initial conditions like v(0) and s(0) provide specific values to solve for these constants, ensuring the solution matches the physical scenario described.
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