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 8.7.9g
Capitolo 8, Problema 8.7.9g

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from 0 to π of sin(t) dt

Guida verificata passo dopo passo
1
Identify the integral to approximate: \(\int_0^{\pi} \sin(t) \, dt\) and note that \(n=4\) subintervals will be used for Simpson's Rule.
Calculate the step size \(h\) using the formula \(h = \frac{b - a}{n} = \frac{\pi - 0}{4} = \frac{\pi}{4}\).
Determine the partition points: \(t_0 = 0\), \(t_1 = \frac{\pi}{4}\), \(t_2 = \frac{\pi}{2}\), \(t_3 = \frac{3\pi}{4}\), and \(t_4 = \pi\).
Apply Simpson's Rule formula: \(S_n = \frac{h}{3} \left[f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + f(t_4)\right]\), where \(f(t) = \sin(t)\).
To find the error bound \(|E_S|\), use the formula for Simpson's Rule error bound: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq t \leq b} |f^{(4)}(t)|\). Calculate the fourth derivative of \(f(t) = \sin(t)\), find its maximum absolute value on \([0, \pi]\), and substitute all values to estimate the error bound.

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:
7m

Concetti chiave

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

Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the function values. It generally provides more accurate results than the Midpoint or Trapezoidal Rules for smooth functions. The formula combines function values at equally spaced points with specific weights to estimate the integral.
Video consigliato:
5:50
Power Rules

Error Bound for Simpson's Rule

The error bound for Simpson's Rule estimates the maximum possible difference between the true integral and the approximation. It depends on the fourth derivative of the integrand, the interval length, and the number of subintervals. Specifically, the error bound helps assess the accuracy of the approximation and ensures the estimate is within a known tolerance.
Video consigliato:
Percorso guidato
04:57
Determining Error and Relative Error

Definite Integral of sin(t) from 0 to π

The integral of sin(t) from 0 to π represents the area under one half-wave of the sine curve, which equals 2 exactly. Understanding this integral provides a benchmark to compare numerical approximations. Since sin(t) is smooth and well-behaved on [0, π], it is suitable for applying Simpson's Rule and analyzing error bounds.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from -2 to 0 of (x² - 1) dx

22
views
Domanda del libro di testo

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

f. u = arccos x

What is the value of the integral?

31
views
Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from 1 to 3 of (2x - 1) dx

21
views
Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from 1 to 2 of 1 / s² ds

13
views
Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from 1 to 2 of x dx

20
views
Domanda del libro di testo

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from 0 to 2 of (t³ + t) dt

26
views