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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.7.9h
Chapter 8, Problem 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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

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

Textbook Question

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

Textbook Question

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

Textbook Question

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

Textbook Question

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

Textbook Question

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