Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.7.20b
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
Verified step by step guidance1
Identify the integral to approximate: \(\int_0^3 \frac{1}{\sqrt{x + 1}} \, dx\).
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 \(f^{(4)}(x)\) of the function \(f(x) = \frac{1}{\sqrt{x + 1}} = (x + 1)^{-1/2}\).
Find the maximum absolute value of \(f^{(4)}(x)\) on the interval \([0, 3]\) by evaluating or estimating \(|f^{(4)}(x)|\) over this range.
Set the error bound \(\frac{(3 - 0)^5}{180 n^4} \max |f^{(4)}(x)| < 10^{-4}\) and solve this inequality for \(n\) to find the minimum even 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 quadratic polynomials to the function. 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) / (180 n^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 Fourth Derivative and Choosing n
To estimate the minimum number of subintervals n, one must find an upper bound K for the absolute value of the fourth derivative of the integrand on the interval. Then, solve the error inequality for n to ensure the error is below the desired tolerance, such as 10^-4.
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07:59Estimating the Area Under a Curve Using Left Endpoints
Related Practice
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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Textbook Question
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
Textbook Question
89. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / x², y = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (i) about the x-axis.
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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.