Skip to main content
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 8.7.16a
Capitolo 8, Problema 8.7.16a

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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (t³ + 1) dt

Guida verificata passo dopo passo
1
Identify the function to be integrated: \(f(t) = t^{3} + 1\) over the interval \([-1, 1]\).
Recall the error bound formula for the Trapezoidal Rule: \(|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
Compute the second derivative of the function: \(f''(t) = \frac{d^{2}}{dt^{2}}(t^{3} + 1) = 6t\).
Determine the maximum absolute value of \(f''(t)\) on the interval \([-1, 1]\): \(\max_{-1 \leq t \leq 1} |6t|\).
Set the error bound less than \(10^{-4}\) and solve the inequality \(\frac{(1 - (-1))^{3}}{12 n^{2}} \max |f''(t)| < 10^{-4}\) for \(n\) to find the minimum number of subintervals needed.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
9m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the integration interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoid areas estimates the integral, with accuracy improving as the number of subintervals increases.
Video consigliato:
5:50
Power Rules

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the second derivative of the integrand. Specifically, the error magnitude is at most (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps estimate how many subintervals are needed to achieve a desired accuracy.
Video consigliato:
Percorso guidato
04:57
Determining Error and Relative Error

Second Derivative and Its Role in Error Estimation

The second derivative of the function indicates the concavity and affects the accuracy of the Trapezoidal Rule. A larger maximum second derivative on the interval means greater potential error, requiring more subintervals for a precise approximation. Calculating or bounding this derivative is essential for applying the error formula.
Video consigliato:
06:02
The Second Derivative Test: Finding Local Extrema
Pratica correlata
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 2 to 4 of 1/(s - 1)² ds

30
views
Domanda del libro di testo

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

20
views
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 3 of (2x - 1) dx

27
views
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 2 of (t³ + t) dt

25
views
Domanda del libro di testo

Finding area

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

a. π/2 ≤ x ≤ 3π/2.

58
views
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 3 of 1/√(x + 1) dx

26
views