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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.7.11b
Chapter 8, Problem 8.7.11b

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 2 of x dx

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1
Identify the function to be integrated: here, \( f(x) = x \) over the interval \([1, 2]\).
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 (must be even), and \( f^{(4)}(x) \) is the fourth derivative of \( f(x) \).
Compute the fourth derivative of \( f(x) = x \). Since \( f(x) = x \), \( f'(x) = 1 \), \( f''(x) = 0 \), and all higher derivatives including \( f^{(4)}(x) = 0 \).
Since \( f^{(4)}(x) = 0 \), the error bound formula implies the error is zero for any \( n \), meaning Simpson's Rule will give the exact value for this integral regardless of the number of subintervals.
Therefore, the minimum number of subintervals needed to achieve an error less than \( 10^{-4} \) is the smallest even number, which is \( n = 2 \), but in fact, any \( n \) will suffice.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It uses an even number of subintervals and generally provides more accurate results than the trapezoidal rule for smooth functions.
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Power Rules

Error Bound for Simpson's Rule

The error bound for Simpson's Rule depends on the fourth derivative of the integrand. Specifically, the error magnitude is less than (K(b - a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.
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Determining Error and Relative Error

Estimating Number of Subintervals

To ensure the approximation error is below a desired threshold, solve the error bound inequality for n, the number of subintervals. This involves finding the maximum of the fourth derivative, substituting into the error formula, and calculating the minimum n that satisfies the error requirement.
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Estimating the Area Under a Curve Using Left Endpoints
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 (b) Simpson’s 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

90. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / √x, y = 0, x = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (i) about the x-axis

1
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Textbook Question

Consider the region bounded by the graphs of

y = arctan(x), y = 0, and x = 1.

b. Find the volume of the solid formed by revolving this region about the y-axis.

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.

1
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Textbook Question

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.

1
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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 (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