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 8.7.21a
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
Verified step by step guidance1
Identify the function to be integrated: \(f(x) = \sin(x + 1)\) over the interval \([0, 2]\).
Recall the error bound formula for the Trapezoidal Rule: the error \(E_T\) satisfies
\[E_T \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|,\]
where \(n\) is the number of subintervals, and \([a, b]\) is the interval of integration.
Compute the second derivative of the function:
\[f'(x) = \cos(x + 1),\]
\[f''(x) = -\sin(x + 1).\]
Determine the maximum absolute value of \(f''(x)\) on \([0, 2]\):
Since \(|\sin(\theta)| \leq 1\) for all real \(\theta\), the maximum of \(|f''(x)| = | -\sin(x + 1)| = |\sin(x + 1)|\) is at most 1 on the interval.
Set the error bound less than \(10^{-4}\) and solve for \(n\):
\[\frac{(2 - 0)^3}{12 n^2} \times 1 < 10^{-4} \implies \frac{8}{12 n^2} < 10^{-4} \implies n^2 > \frac{8}{12 \times 10^{-4}}.\]
From here, solve 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 integration interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.
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Power Rules
Error Bound for the Trapezoidal Rule
The error bound for the Trapezoidal Rule depends on the second derivative of the integrand. Specifically, the error magnitude is at most (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps determine how many subintervals are needed to achieve a desired accuracy.
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04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the function indicates the concavity and affects the curvature of the graph. In error estimation for the Trapezoidal Rule, the maximum absolute value of the second derivative over the interval controls the error size. Calculating or bounding this value is essential to apply the error formula and find the minimum n.
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06:02The Second Derivative Test: Finding Local Extrema
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 2 to 4 of 1/(s - 1)² ds
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.
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 -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