Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
b. Show that the inner and outer surfaces of the solid have infinite area.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.7.13b
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
Verified step by step guidance1
Identify the function to be integrated: \(f(x) = x^2 + 1\) over the interval \([-1, 1]\).
Recall the error bound formula for Simpson's Rule: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\), where \(n\) is the number of subintervals (and must be even).
Compute the fourth derivative of the function \(f(x)\). Since \(f(x) = x^2 + 1\), find \(f^{(4)}(x)\).
Determine the maximum value of \(|f^{(4)}(x)|\) on the interval \([-1, 1]\).
Set the error bound \(|E_S|\) less than \(10^{-4}\) and solve the inequality for \(n\) to find the minimum number of subintervals needed.
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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 trapezoidal or midpoint rules for smooth functions.
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Power Rules
Error Bound for Simpson's Rule
The error bound for Simpson's Rule depends on the fourth derivative of the integrand and the number of subintervals. Specifically, the error magnitude is less than (K(b−a)^5)/(180n^4), where K bounds the absolute value of the fourth derivative on [a,b], and n is the number of subintervals.
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04:57Determining Error and Relative Error
Estimating the Number of Subintervals
To ensure the approximation error is below a desired threshold, solve the error bound inequality for n, the number of subintervals. This involves finding the maximum of the fourth derivative on the interval and rearranging the error formula to find the minimum n that satisfies the error tolerance.
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07:59Estimating the Area Under a Curve Using Left Endpoints
Related Practice
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of sin(x + 1) dx
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
b. π ≤ x ≤ 2π.
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Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 1) dx