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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 8.7.9g
Chapter 8, Problem 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

Verified step by step guidance
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.

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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. The formula combines function values at equally spaced points with specific weights to estimate the integral.
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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.
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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.
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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 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 -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 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 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

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

∫ from 1 to 2 of x 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 2 of (t³ + t) dt