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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 8.7.9h
Capitolo 8, Problema 8.7.9h

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

Guida verificata passo dopo passo
1
First, write down the integral to be evaluated: \(\int_0^{\pi} \sin(t) \, dt\).
Recall that the exact value of the integral can be found by finding an antiderivative of \(\sin(t)\), which is \(-\cos(t)\).
Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits: calculate \(-\cos(\pi)\) and \(-\cos(0)\).
Subtract the value at the lower limit from the value at the upper limit to find the exact value of the integral: \(-\cos(\pi) - (-\cos(0))\).
To find the Simpson's Rule error bound \(|E_S|\), use the formula \(|E_S| \leq \frac{K(b-a)^5}{180n^4}\), where \(K\) is the maximum absolute value of the fourth derivative of \(f(t) = \sin(t)\) on \([0, \pi]\), \(a=0\), \(b=\pi\), and \(n\) is the number of subintervals used in Simpson's Rule.

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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.
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Power Rules

Exact Evaluation of Definite Integrals

Exact evaluation involves finding the precise value of an integral using antiderivatives and the Fundamental Theorem of Calculus. For ∫₀^π sin(t) dt, the antiderivative of sin(t) is -cos(t), which is evaluated at the bounds to find the exact integral value.
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Definition of the Definite Integral

Error Estimation in Numerical Integration (|ES|)

|ES| represents the absolute error in Simpson's Rule approximation, calculated as the absolute difference between the exact integral value and the Simpson's Rule estimate. Understanding error estimation helps assess the accuracy of numerical methods.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
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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 1 / s² ds

22
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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

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