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Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.7.86a
Capitolo 8, Problema 8.7.86a

Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
a. Find an expression for L.

Guida verificata passo dopo passo
1
Step 1: Recall the formula for the arc length of a curve y = f(x) from x = a to x = b: \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \). This formula will be used to find the arc length of the parabola.
Step 2: Identify the function \( f(x) = x^2 \) and compute its derivative \( \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x \). Substitute \( \frac{dy}{dx} \) into the arc length formula.
Step 3: Substitute \( \frac{dy}{dx} = 2x \) into the formula \( L = \int_0^c \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \), resulting in \( L = \int_0^c \sqrt{1 + (2x)^2} \, dx \). Simplify the expression inside the square root.
Step 4: Simplify \( \sqrt{1 + (2x)^2} \) to \( \sqrt{1 + 4x^2} \). The arc length formula now becomes \( L = \int_0^c \sqrt{1 + 4x^2} \, dx \). This is the expression for the arc length.
Step 5: Note that the integral \( \int_0^c \sqrt{1 + 4x^2} \, dx \) may require advanced techniques such as substitution or numerical methods to evaluate for specific values of \( c \). The expression \( L = \int_0^c \sqrt{1 + 4x^2} \, dx \) is the general formula for the arc length of the parabola.

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Concetti chiave

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Arc Length Formula

The arc length of a curve defined by a function y = f(x) from x = a to x = b is given by the formula L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula derives from the Pythagorean theorem, where the infinitesimal segments of the curve are approximated as straight lines.
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Arc Length of Parametric Curves

Derivative of a Function

The derivative of a function, denoted as dy/dx, represents the rate of change of the function with respect to x. For the parabola f(x) = x², the derivative is f'(x) = 2x, which is essential for calculating the arc length as it appears in the arc length formula.
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Derivatives of Other Trig Functions

Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of finding the arc length, the definite integral is used to sum the lengths of infinitesimal segments of the curve from x = 0 to x = c, providing the total length L(c).
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Definition of the Definite Integral
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