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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 6.1.63
Chapter 6, Problem 6.1.63

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t) = t(25−t²)^1/2, for 0≤t≤5

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Identify the time interval over which the velocity function is defined, which is from \(t=0\) to \(t=5\) seconds.
Recall that the total distance traveled over the time interval can be found by integrating the velocity function \(v(t)\) with respect to time: \(\text{Distance} = \int_0^5 v(t) \, dt\).
Set up the integral for the given velocity function: \(\int_0^5 t \sqrt{25 - t^2} \, dt\).
Calculate the definite integral to find the total distance traveled during the time interval. (You can use substitution or other integration techniques to solve this integral.)
Once the total distance is found, find the equivalent constant velocity \(v_c\) by dividing the total distance by the total time interval: \(v_c = \frac{\text{Distance}}{5}\).

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Key Concepts

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

Average Velocity

Average velocity over a time interval is defined as the total displacement divided by the total time. It represents a constant velocity that would cover the same distance in the same time period, making it essential for comparing variable velocity functions to a constant velocity.
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Average Value of a Function

Definite Integral for Displacement

The definite integral of a velocity function over a time interval gives the total displacement traveled during that period. Calculating this integral is necessary to find the total distance covered, which is then used to determine the equivalent constant velocity.
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Definition of the Definite Integral

Velocity Function and Domain

Understanding the given velocity function v(t) = t(25−t²)^(1/2) and its domain 0 ≤ t ≤ 5 is crucial. This function describes how velocity changes over time, and recognizing its behavior helps in setting up the integral and interpreting the physical meaning of the problem.
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Using The Velocity Function
Related Practice
Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


{Use of Tech} y² = ln x,y² = ln x³, and y=2; about the x-axis

Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=3x+4, for 0≤x≤6; about the x-axis

Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.

x=4y^3/2−y^1/2 / 12, for 1≤y≤4; about the y-axis

Textbook Question

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.


ρ(x) = {x² if 0≤x≤1 {x(2-x) if 1<x≤2

Textbook Question

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]

Textbook Question

The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.