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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 8.7.42
Capitolo 8, Problema 8.7.42

41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.

Guida verificata passo dopo passo
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Step 1: Recall the formula for the length of a curve. The length of a curve y = f(x) from x = a to x = b is given by: ∫abdydx2+1dx. This formula accounts for both the vertical and horizontal changes along the curve.
Step 2: Compute the derivative of y = x^(3/2) + 8 with respect to x. The derivative is: dydx=32x12. The constant 8 disappears because its derivative is 0.
Step 3: Substitute the derivative into the curve length formula. The integrand becomes: 32x122+1. Simplify the square of the derivative term.
Step 4: Simplify the integrand further. The square of the derivative term becomes: 94x. Thus, the integrand is: 94x+1.
Step 5: Use a table of integrals to evaluate the integral of 94x+1 from x = 0 to x = 2. Look for a matching formula in the table of integrals and apply it to compute the curve length.

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Arc Length Formula

The arc length formula is used to calculate the length of a curve defined by a function. For a function y = f(x), the length L from x = a to x = b is given by the integral L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula incorporates the derivative of the function to account for the slope of the curve, providing an accurate measure of its length.
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Arc Length of Parametric Curves

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of the arc length formula, the derivative dy/dx is crucial as it helps determine how steep the curve is at any point, affecting the overall length calculation.
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Finding Differentials

Definite Integrals

Definite integrals are used to calculate the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of finding arc length, the definite integral computes the total length of the curve between two points by summing infinitesimally small segments of the curve, providing a precise measurement of its extent.
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Definition of the Definite Integral
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