Domanda del libro di testo
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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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.7.13a
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 -1 to 1 of (x² + 1) dx
Guida verificata passo dopo passo1
Identify the function to be integrated: \(f(x) = x^{2} + 1\) over the interval \([-1, 1]\).
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 second derivative of the function: \(f''(x) = \frac{d^{2}}{dx^{2}}(x^{2} + 1) = 2\).
Determine the maximum value of \(|f''(x)|\) on the interval \([-1, 1]\), which is \(2\) since \(f''(x)\) is constant.
Set up the inequality for the error bound to be less than \(10^{-4}\) and solve for \(n\): \(\frac{(1 - (-1))^{3}}{12 n^{2}} \times 2 < 10^{-4}\). Simplify and solve for \(n\) to find the minimum number of subintervals needed.
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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 sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.
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5:50
Power Rules
Error Bound for the Trapezoidal Rule
The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. 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 estimate how many subintervals are needed for a desired accuracy.
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Percorso guidato
04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the integrand measures the function's concavity and affects the error in the Trapezoidal Rule. A larger maximum second derivative on the interval implies a larger potential error, requiring more subintervals to achieve a given accuracy. Calculating or bounding this derivative is essential for applying the error formula.
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06:02The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo
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.
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Domanda del libro di testo
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
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Domanda del libro di testo
Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
a. u = 1/(x + 1)
What is the value of the integral?
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Domanda del libro di testo
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
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Domanda del libro di testo
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 1 to 2 of 1/s² ds
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