Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.
a. About the y-axis.
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 8.7.18a
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 4 of 1/(s - 1)² ds
Verified step by step guidance1
Identify the function to be integrated: \(f(s) = \frac{1}{(s - 1)^2}\) over the interval \([2, 4]\).
Recall the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
Compute the first derivative \(f'(s)\) and then the second derivative \(f''(s)\) of the function \(f(s)\).
Determine the maximum absolute value of \(f''(s)\) on the interval \([2, 4]\) by analyzing \(f''(s)\) or evaluating it at critical points and endpoints.
Set the error bound \(\frac{(4 - 2)^3}{12 n^2} \max |f''(s)| < 10^{-4}\) and solve this 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.
Trapezoidal Rule
The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The accuracy depends on the number of subintervals; more subintervals generally yield better approximations.
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Error Bound for the Trapezoidal Rule
The error bound for the Trapezoidal Rule estimates the maximum possible error in the approximation. It depends on the second derivative of the function, the length of the interval, and the number of subintervals. Specifically, the error is bounded by (b−a)³/(12n²) times the maximum of |f''(x)| on [a,b].
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Second Derivative and Its Role in Error Estimation
The second derivative of the integrand measures the concavity of the function and influences the error in the Trapezoidal Rule. To estimate the error bound, one must find the maximum absolute value of the second derivative on the interval, which helps determine how many subintervals are needed for a desired accuracy.
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Related Practice
Textbook Question
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
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
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 3 of 1/√(x + 1) dx
Textbook Question
Centroid:
Find the centroid of the region cut from the first quadrant by the curve
y = 1/√(x + 1) and the line x = 3.