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

Guida verificata passo dopo passo

Identify the integral to approximate: \(\int_{-2}^{0} (x^{2} - 1) \, dx\) and note that \(n = 4\) subintervals will be used for Simpson's Rule.

Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = -2\) and \(b = 0\).

Determine the \(x\)-values (nodes) at which the function will be evaluated: \(x_0 = -2\), \(x_1 = -2 + \Delta x\), \(x_2 = -2 + 2\Delta x\), \(x_3 = -2 + 3\Delta x\), and \(x_4 = 0\).

Apply Simpson's Rule formula: \(S_n = \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)\right]\) where \(f(x) = x^{2} - 1\).

To find the upper bound for the error \(|E_S|\), use the error bound formula for Simpson's Rule: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\) Calculate the fourth derivative \(f^{(4)}(x)\) of the function and find its maximum absolute value on the interval \([-2, 0]\).

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:

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 function over the interval and the number of subintervals n. Specifically, the error bound is proportional to the maximum absolute value of the fourth derivative and inversely proportional to n to the fourth power, ensuring accuracy improves with more steps.

Video consigliato:

Percorso guidato

04:57

Determining Error and Relative Error

Definite Integral and Function Behavior

Understanding the definite integral involves interpreting the area under the curve of the function between given limits. For the function f(x) = x² - 1 on [-2, 0], knowing its shape and derivatives helps in applying numerical methods and error analysis. Recognizing the function's smoothness and polynomial nature simplifies calculating derivatives needed for error bounds.

Video consigliato:

Percorso guidato

05:43

Definition of the Definite Integral

Pratica correlata

Domanda del libro di testo