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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.71a
Chapter 8, Problem 8.8.71a

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
a. Find a Simpson's Rule approximation to the integral from 1 to 2 of √(sin x) dx using n = 20 subintervals.

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Step 1: Understand the problem. We are tasked with approximating the integral of f(x) = √(sin x) over the interval [1, 2] using Simpson's Rule with n = 20 subintervals. Simpson's Rule is a numerical integration method that uses parabolic approximations to estimate the area under a curve.
Step 2: Recall the formula for Simpson's Rule: \( S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 4f(x_{n-1}) + f(x_n) \right] \), where \( \Delta x = \frac{b-a}{n} \), \( x_0, x_1, \dots, x_n \) are the subinterval points, and \( n \) must be even.
Step 3: Calculate \( \Delta x \). Since \( a = 1 \), \( b = 2 \), and \( n = 20 \), \( \Delta x = \frac{2-1}{20} = 0.05 \). The subinterval points are \( x_0 = 1, x_1 = 1.05, x_2 = 1.1, \dots, x_{20} = 2 \).
Step 4: Evaluate \( f(x) = \sqrt{\sin x} \) at each subinterval point \( x_0, x_1, \dots, x_{20} \). For example, \( f(x_0) = \sqrt{\sin(1)} \), \( f(x_1) = \sqrt{\sin(1.05)} \), and so on. These values will be used in the Simpson's Rule formula.
Step 5: Apply the Simpson's Rule formula. Substitute \( \Delta x = 0.05 \), the calculated \( f(x_i) \) values, and the coefficients (1 for \( f(x_0) \) and \( f(x_n) \), 4 for odd-indexed terms, and 2 for even-indexed terms) into the formula. Simplify the expression to approximate the integral.

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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 the definite integral of a function. It uses parabolic segments to estimate the area under a curve, providing a more accurate approximation than methods like the Trapezoidal Rule. The formula involves evaluating the function at evenly spaced points and applying weights to these values, which is particularly useful for functions that are smooth and continuous.
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Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, over the interval.
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Error Estimation in Numerical Integration

Error estimation in numerical integration involves determining how closely a numerical approximation, like Simpson's Rule, approximates the actual value of the integral. Theorem 8.1 likely provides a formula or method to estimate this error based on the function's behavior and the number of subintervals used. Understanding error estimation is crucial for assessing the reliability of numerical results.
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Related Practice
Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

68. Let f(x) = e^(x²).

a. Find a Trapezoid Rule approximation to ∫[0 to 1] e^(x²) dx using n = 50 subintervals.

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Textbook Question

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

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Textbook Question

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a. Find the probability that the computer chip fails after 15,000 hr of operation.

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Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

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