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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.7.11b
Capitolo 8, Problema 8.7.11b

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of x dx

Guida verificata passo dopo passo
1
Identify the function to be integrated: here, \( f(x) = x \) over the interval \([1, 2]\).
Recall the error bound formula for Simpson's Rule: \[ |E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)| \], where \( n \) is the number of subintervals (must be even), and \( f^{(4)}(x) \) is the fourth derivative of \( f(x) \).
Compute the fourth derivative of \( f(x) = x \). Since \( f(x) = x \), \( f'(x) = 1 \), \( f''(x) = 0 \), and all higher derivatives including \( f^{(4)}(x) = 0 \).
Since \( f^{(4)}(x) = 0 \), the error bound formula implies the error is zero for any \( n \), meaning Simpson's Rule will give the exact value for this integral regardless of the number of subintervals.
Therefore, the minimum number of subintervals needed to achieve an error less than \( 10^{-4} \) is the smallest even number, which is \( n = 2 \), but in fact, any \( n \) will suffice.

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Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It uses an even number of subintervals and generally provides more accurate results than the trapezoidal rule for smooth functions.
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Power Rules

Error Bound for Simpson's Rule

The error bound for Simpson's Rule depends on the fourth derivative of the integrand. Specifically, the error magnitude is less than (K(b - a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.
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Percorso guidato
04:57
Determining Error and Relative Error

Estimating Number of Subintervals

To ensure the approximation error is below a desired threshold, solve the error bound inequality for n, the number of subintervals. This involves finding the maximum of the fourth derivative, substituting into the error formula, and calculating the minimum n that satisfies the error requirement.
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Percorso guidato
07:59
Estimating the Area Under a Curve Using Left Endpoints
Pratica correlata
Domanda del libro di testo

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

b. Show that the inner and outer surfaces of the solid have infinite area.

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Domanda del libro di testo

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 2 of sin(x + 1) dx

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Domanda del libro di testo

Finding area

Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:

c. 2π ≤ x ≤ 3π.

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Domanda del libro di testo

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

30
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Domanda del libro di testo

89. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / x², y = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

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Domanda del libro di testo

90. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / √x, y = 0, x = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

25
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