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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.5.33c
Capitolo 6, Problema 6.5.33c

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


Arc length may be negative if f(x) < 0 on part of the interval in question.

Guida verificata passo dopo passo
1
Recall the definition of arc length for a function \(f(x)\) on an interval \([a,b]\): the arc length \(L\) is given by the integral \(L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\).
Note that the integrand \(\sqrt{1 + (f'(x))^2}\) is always non-negative because it is a square root of a sum of squares, which cannot be negative.
Since the integrand is non-negative and the limits of integration satisfy \(a < b\), the value of the integral (arc length) must be non-negative.
The value of \(f(x)\) itself (whether positive or negative) does not affect the sign of the arc length because the formula depends on \(f'(x)\), the derivative, and the square root expression, not directly on \(f(x)\).
Therefore, arc length cannot be negative even if \(f(x) < 0\) on part of the interval; the statement is false.

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Definition of Arc Length

Arc length measures the distance along a curve between two points and is always a non-negative quantity. It is calculated by integrating the square root of 1 plus the derivative squared, ensuring the result represents a length, which cannot be negative.
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06:29
Arc Length of Parametric Curves

Role of the Function's Sign in Arc Length

The sign of the function f(x) does not affect the arc length because arc length depends on the magnitude of the derivative, not the function's value. Even if f(x) is negative, the length along the curve remains positive.
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06:29
Arc Length of Parametric Curves

Integral of Absolute Values and Non-negativity

Arc length involves integrating the square root of the sum of squares, which is always non-negative. This ensures the integral accumulates positive values, preventing the arc length from being negative regardless of the function's behavior.
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05:59
Initial Value Problems Example 2
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Domanda del libro di testo

Determine whether the following statements are true and give an explanation or counterexample. 


c. Let f(x)=12x^2. The area of the surface generated when the graph of f on [−4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis. 

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Domanda del libro di testo

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

c. Write an integral for the volume of the solid using the shell method.

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Domanda del libro di testo

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

c. Write an integral for the volume of the solid using the shell method.

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Domanda del libro di testo

{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. 


c. At what times does the mass reach its low point the first three times? 

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Domanda del libro di testo

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

c. The Spokane River flows out of Lake Coeur d’Alene, which contains approximately 0.67mi³ of water. Determine the percentage of Lake Coeur d’Alene’s volume that flows through Spokane in April and June.

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Domanda del libro di testo

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

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