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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 8.7.7h
Capitolo 8, Problema 8.7.7h

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
b. Evaluate the integral directly and find |ES|.
∫ from 1 to 2 of 1 / s² ds

Guida verificata passo dopo passo
1
Identify the integral to evaluate: \(\int_{1}^{2} \frac{1}{s^{2}} \, ds\).
Rewrite the integrand \(\frac{1}{s^{2}}\) as \(s^{-2}\) to make integration straightforward.
Find the antiderivative of \(s^{-2}\). Recall that the integral of \(s^{n}\) with respect to \(s\) is \(\frac{s^{n+1}}{n+1} + C\) for \(n \neq -1\).
Apply the antiderivative formula to \(s^{-2}\), which gives \(-s^{-1} + C\).
Evaluate the definite integral by substituting the limits 1 and 2 into the antiderivative: compute \([-s^{-1}]_{1}^{2} = (-\frac{1}{2}) - (-1)\).

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Definite Integral

A definite integral calculates the exact area under a curve between two limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the interval bounds. Evaluating this integral involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral

Simpson's Rule

Simpson's Rule is a numerical method to approximate definite integrals by fitting parabolas through segments of the function. It generally provides more accurate results than the Midpoint or Trapezoidal Rules, especially for smooth functions. The interval is divided into an even number of subintervals for the approximation.
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Power Rules

Error Bound for Simpson's Rule (|ES|)

The error bound |ES| estimates the maximum possible difference between the exact integral and Simpson's Rule approximation. It depends on the fourth derivative of the function and the width of the subintervals. Calculating |ES| helps assess the accuracy of the numerical approximation.
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Intro to the Chain Rule Example 1
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

b. Evaluate the integral directly and find |ES|.

∫ from 0 to π of sin(t) dt

38
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

b. Evaluate the integral directly and find |ES|.

∫ from -2 to 0 of (x² - 1) dx

22
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 -2 to 0 of (x² - 1) dx

22
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

b. Evaluate the integral directly and find |ES|.

∫ from 1 to 3 of (2x - 1) dx

29
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

b. Evaluate the integral directly and find |ES|.

∫ from 1 to 2 of x dx

24
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