Domanda del libro di testo
4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
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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 8.7.20a
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
Guida verificata passo dopo passo1
Identify the function to be integrated: \(f(x) = \frac{1}{\sqrt{x + 1}}\) over the interval \([0, 3]\).
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 \(f'(x)\) and then the second derivative \(f''(x)\) of the function \(f(x)\).
Find the maximum absolute value of \(f''(x)\) on the interval \([0, 3]\) by analyzing \(f''(x)\) or evaluating it at critical points and endpoints.
Set the error bound \(\frac{(3 - 0)^3}{12 n^2} \max |f''(x)| < 10^{-4}\) and solve this inequality for \(n\) to estimate 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 accuracy depends on the number of subintervals; more subintervals generally yield better approximations.
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Power Rules
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 integrand and the number of subintervals. Specifically, the error is bounded by (b−a)^3/(12n^2) times the maximum of |f''(x)| on [a,b].
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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 function indicates the concavity and affects 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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06:02The Second Derivative Test: Finding Local Extrema
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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.)
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