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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 8.8.66a
Chapter 8, Problem 8.8.66a

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.

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Identify the interval of integration and the number of subintervals: here, the integral is from \(-1\) to \(1\), and \(n = 30\) subintervals.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = -1\) and \(b = 1\).
Determine the midpoints of each subinterval. For the \(i^{th}\) subinterval, the midpoint \(x_i^*\) is given by \(x_i^* = a + \left(i - \frac{1}{2}\right) \Delta x\) for \(i = 1, 2, ..., n\).
Evaluate the function \(f(x) = \cos(x^2)\) at each midpoint \(x_i^*\) to get \(f(x_i^*)\).
Approximate the integral using the Midpoint Rule formula: \(M_n = \Delta x \sum_{i=1}^n f(x_i^*)\). This sum multiplied by the subinterval width gives the approximation.

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Key Concepts

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

Midpoint Rule for Numerical Integration

The Midpoint Rule approximates the definite integral of a function by dividing the interval into equal subintervals and using the function's value at each subinterval's midpoint to estimate the area. This method improves accuracy over using endpoints and is especially useful when the function is difficult to integrate analytically.
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Additional Rules for Indefinite Integrals

Partitioning the Interval into Subintervals

To apply numerical methods like the Midpoint Rule, the integration interval [a, b] is divided into n equal subintervals, each of width Δx = (b - a)/n. The midpoints of these subintervals are then used to evaluate the function, which forms the basis for the approximation.
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Interval of Convergence

Error Estimation in Numerical Integration

Error estimation provides a bound on the difference between the exact integral and its numerical approximation. Theorem 8.1 likely gives an error bound for the Midpoint Rule, which depends on the second derivative of the function and the number of subintervals, helping assess the accuracy of the approximation.
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Determining Error and Relative Error
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

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. If x = 4 tanθ, then cscθ = 4/x.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.

Textbook Question

62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.

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

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Textbook Question

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.

Textbook Question

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

48. ∫ sin(3x) cos⁶(3x) dx