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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 8.7.7h
Chapter 8, Problem 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

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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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Key Concepts

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

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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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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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

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

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

∫ from 1 to 2 of x dx