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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 6.5.35a
Capitolo 6, Problema 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

Guida verificata passo dopo passo
1
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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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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Arc Length of Parametric Curves

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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Finding Area Between Curves on a Given Interval

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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Integrals of General Exponential Functions
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