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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 6.1.62
Chapter 6, Problem 6.1.62

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)=2 sin t, for 0≤t≤π

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Identify the given velocity function: \(v(t) = 2 \sin t\) over the interval \(0 \leq t \leq \pi\).
Recall that the distance traveled over a time interval is the integral of the velocity function over that interval. So, calculate the total distance by evaluating the definite integral \(\int_0^{\pi} 2 \sin t \, dt\).
Compute the integral: Use the antiderivative of \(\sin t\), which is \(-\cos t\), to find \(\int 2 \sin t \, dt = -2 \cos t + C\).
Evaluate the definite integral from \(0\) to \(\pi\): Calculate \([-2 \cos t]_0^{\pi} = (-2 \cos \pi) - (-2 \cos 0)\).
To find the equivalent constant velocity \(v_c\), divide the total distance traveled by the total time interval length \(\pi\): \(v_c = \frac{\int_0^{\pi} 2 \sin t \, dt}{\pi}\).

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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 as the varying velocity function.
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Definite Integral and Displacement

The definite integral of a velocity function over a time interval gives the total displacement during that period. Calculating the integral of v(t) = 2 sin t from 0 to π provides the total distance traveled.
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Definition of the Definite Integral

Relationship Between Velocity and Distance

Velocity is the rate of change of position. To find an equivalent constant velocity, one must use the total distance traveled and divide it by the total time, ensuring the constant velocity covers the same displacement as the original function.
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Related Practice
Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=x^2−2x+1 and y=5x−9

Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.


Textbook Question

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


y=(3x)^1/3 , for 0≤x≤8/3; about the y-axis

Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.


y = 2

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} y = √50 -2x², in the first quadrant; about the x-axis