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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 6.5.33c
Chapter 6, Problem 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.

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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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Key Concepts

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

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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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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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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Initial Value Problems Example 2
Related Practice
Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


c. When do they meet? How far has each person traveled when they meet?

Textbook Question

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.


c. How far has the cyclist traveled when her velocity is 250 m/min?

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Textbook Question

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

c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

Textbook Question

Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 


c. Find the minimum decay constant k for which the total oil reserves will last forever.

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Textbook Question

Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.


c. Find the distance traveled by the object on the interval 0≤t≤6.

Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

c. The position at t=5