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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 6.5.35a
Chapter 6, Problem 6.5.35a

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
a. ∫a^b √1+16x⁴ dx

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Recall that the arc length of a function \(y = f(x)\) on the interval \([a, b]\) is given by the integral \(\int_a^b \sqrt{1 + (f'(x))^2} \, dx\).
Compare the given integral \(\int_a^b \sqrt{1 + 16x^4} \, dx\) with the arc length formula. This means that \(\sqrt{1 + (f'(x))^2} = \sqrt{1 + 16x^4}\).
From the equality inside the square roots, deduce that \((f'(x))^2 = 16x^4\).
Take the square root of both sides to find \(f'(x) = \pm 4x^2\).
Integrate \(f'(x)\) to find the family of functions: \(f(x) = \pm \int 4x^2 \, dx = \pm \frac{4}{3} x^3 + C\), where \(C\) is an arbitrary constant.

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Key Concepts

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

Arc Length Formula

The arc length of a differentiable function y = f(x) over [a, b] is given by the integral ∫_a^b √(1 + (f'(x))²) dx. This formula measures the length of the curve by summing infinitesimal line segments, incorporating the slope of the function through its derivative.
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Relationship Between the Integrand and the Derivative

In the arc length integral, the integrand √(1 + (f'(x))²) reveals how the derivative f'(x) relates to the given expression under the square root. To find functions with a specified arc length integral, one must equate (f'(x))² to the expression inside the integral minus 1 and solve for f'(x).
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Family of Functions and Integration

Once f'(x) is determined, integrating it yields a family of functions differing by a constant of integration. This reflects the non-uniqueness of solutions, as any vertical shift of the function preserves the same derivative and thus the same arc length integral.
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Related Practice
Textbook Question

Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).


a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

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Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.


a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.


C′(x) = 300+10x−0.01x²

Textbook Question

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


a. The distance traveled by an object moving along a line is the same as the displacement of the object.

Textbook Question

{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. 


a. Determine the position function, for t≥0.

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. 


a. Find Q(t), the total amount of oil extracted by the nation after t years.

Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.