Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.7.21b

Chapter 8, Problem 8.7.21b

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

Verified step by step guidance

Identify the integral to approximate: \(\int_0^2 \sin(x + 1) \, dx\).

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).

Find the fourth derivative of the integrand \(f(x) = \sin(x + 1)\). Since \(f(x) = \sin(x + 1)\), its derivatives cycle every four steps: \(f^{(1)}(x) = \cos(x + 1)\), \(f^{(2)}(x) = -\sin(x + 1)\), \(f^{(3)}(x) = -\cos(x + 1)\), and \(f^{(4)}(x) = \sin(x + 1)\).

Determine the maximum value of \(|f^{(4)}(x)|\) on the interval \([0, 2]\). Since \(f^{(4)}(x) = \sin(x + 1)\), find the maximum absolute value of \(\sin(x + 1)\) for \(x\) in \([0, 2]\).

Set up the inequality for the error bound to be less than \(10^{-4}\): \(\frac{(2 - 0)^5}{180 n^4} \max |f^{(4)}(x)| < 10^{-4}\). Solve this 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 quadratic polynomials to the function. It generally provides more accurate results than the trapezoidal rule for smooth functions.

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.

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, substituting into the error formula, and calculating the minimum n that satisfies the error requirement.

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Related Practice

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

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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

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

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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.

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Textbook Question