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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 8.7.13b
Chapter 8, Problem 8.7.13b

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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1
Identify the function to be integrated: \(f(x) = x^2 + 1\) over the interval \([-1, 1]\).
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 of the function \(f(x)\). Since \(f(x) = x^2 + 1\), find \(f^{(4)}(x)\).
Determine the maximum value of \(|f^{(4)}(x)|\) on the interval \([-1, 1]\).
Set the error bound \(|E_S|\) less than \(10^{-4}\) and solve the inequality 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.

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 parabolas through the function values. It generally provides more accurate results than the trapezoidal or midpoint rules for smooth functions.
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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)/(180n^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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Estimating the 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 on the interval and rearranging the error formula to find the minimum n that satisfies the error tolerance.
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Related Practice
Textbook Question

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

b. Show that the inner and outer surfaces of the solid have infinite area.

1
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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

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

Textbook Question

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

y = 1 / x², y = 0, and x = 1.

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

2
views
Textbook Question

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.