Domanda del libro di testo
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
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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 8.7.20b
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
Guida verificata passo dopo passo1
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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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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5:50
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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Percorso guidato
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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Percorso guidato
07:59Estimating the Area Under a Curve Using Left Endpoints
Pratica correlata
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 0 to 2 of sin(x + 1) dx
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Domanda del libro di testo
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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Domanda del libro di testo
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
b. 3π/2 ≤ x ≤ 5π/2.
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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 and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.
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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 (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
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