Textbook Question
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(c) β«β β· Ζ(π) dπ
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.R.96c
Velocity to displacement An object travels on the π-axis with a velocity given by v(t) = 2t + 5, for 0 β€ t β€ 4.
(c) True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant), for 0 β€ t β€ 4. .
Verified step by step guidance1
Step 1: Understand the problem. The velocity function v(t) = 2t + 5 represents the instantaneous velocity of the object at time t. To determine if the object would travel the same distance at its average velocity, we need to compare the displacement using the instantaneous velocity and the displacement using the average velocity.
Step 2: Calculate the displacement using the instantaneous velocity. Displacement is the integral of velocity over time. Set up the integral: β«[0 to 4] v(t) dt = β«[0 to 4] (2t + 5) dt. Use the rules of integration to compute this integral.
Step 3: Find the average velocity. The average velocity over the interval [0, 4] is given by the formula: v_avg = (1 / (b - a)) β«[a to b] v(t) dt, where a = 0 and b = 4. Substitute the values and compute the average velocity.
Step 4: Calculate the displacement using the average velocity. Displacement at constant velocity is given by: displacement = v_avg Γ time. Multiply the average velocity by the total time interval (4 seconds).
Step 5: Compare the two displacements. If the displacement calculated using the instantaneous velocity matches the displacement calculated using the average velocity, the statement is true. Otherwise, it is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Displacement
Velocity is the rate of change of displacement with respect to time, indicating how fast an object is moving and in which direction. Displacement, on the other hand, is the overall change in position of the object. Understanding the relationship between these two concepts is crucial for analyzing motion, as displacement can be calculated by integrating the velocity function over a given time interval.
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Using The Velocity Function
Average Velocity
Average velocity is defined as the total displacement divided by the total time taken. It represents a constant velocity that would result in the same displacement as the actual varying velocity over a specific time interval. In this context, calculating the average velocity over the interval from t = 0 to t = 4 is essential to determine if the object would travel the same distance as when using the instantaneous velocity function.
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06:37Average Value of a Function
Integration
Integration is a fundamental concept in calculus used to find the area under a curve, which in the context of motion, corresponds to calculating displacement from a velocity function. By integrating the velocity function v(t) = 2t + 5 over the interval [0, 4], one can determine the total displacement of the object. This process is key to answering questions about the distance traveled by the object during the specified time period.
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06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
Area functions and the Fundamental Theorem Consider the function
Ζ(t) = { t if β2 β€ t < 0
tΒ²/2 if 0 β€ t β€ 2
and its graph shown below. Let F(π) = β«ββΛ£ Ζ(t) dt and G(π) = β«ββΛ£ Ζ(t) dt.
(c) Use the Fundamental Theorem to find an expression for F '(π) for 0 β€ π < 2.
Textbook Question
Evaluating integrals Evaluate the following integrals.
β«βΟ/β^Ο/Β² (cos 2π + cos π sin π β 3 sin πβ΅) dπ
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(f) β«βα΅ (2 Ζ(π) β3g (π)) dπ = 2 β«βα΅ Ζ(π) dπ + 3 β«βα΅ g(π) dπ .
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(b) Given an area function A(π) = β«βΛ£ Ζ(t) dt and an antiderivative F of Ζ, it follows that A'(π) = F(π) .
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Textbook Question
Evaluating integrals Evaluate the following integrals.
β« (cos 7Ο) /(16 + sinΒ² 7Ο) dΟ