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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 8.8.63
Chapter 8, Problem 8.8.63

63. (Use of Tech) Normal distribution of heights
The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral
(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.
What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.

Verified step by step guidance
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Step 1: Recognize that the problem involves calculating the area under the curve of a normal distribution between the limits a = 66 and b = 72. The integral provided represents the probability density function for the normal distribution.
Step 2: Rewrite the integral in terms of the given values. The integral becomes (1/(3√(2π))) ∫₆₆⁷² e^(-((x-69)/3)²/2) dx. This represents the fraction of men with heights between 66 and 72 inches.
Step 3: Use numerical integration methods to approximate the value of the integral. Techniques such as the trapezoidal rule or Simpson's rule can be applied. Divide the interval [66, 72] into subintervals and calculate the area for each subinterval.
Step 4: Experiment with the number of subintervals. Start with a small number of subintervals (e.g., 10) and gradually increase the number until successive approximations differ by less than 10⁻³. This ensures the accuracy of the result.
Step 5: Once the integral is approximated, multiply the result by (1/(3√(2π))) to account for the scaling factor of the normal distribution. Convert the final value into a percentage by multiplying by 100.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the heights of U.S. men follow a normal distribution with a mean of 69 inches and a standard deviation of 3 inches.
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Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, between two specified limits. In this problem, the integral represents the area under the normal distribution curve between the heights of 66 and 72 inches. This area corresponds to the probability of randomly selecting a man whose height falls within this range.
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Numerical Integration

Numerical integration is a computational technique used to approximate the value of a definite integral when an analytical solution is difficult or impossible to obtain. Methods such as the trapezoidal rule or Simpson's rule can be employed to divide the area into subintervals, allowing for successive approximations. The goal is to refine these approximations until the difference between them is less than a specified tolerance, in this case, 10⁻³.
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Related Practice
Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

31. ∫ √(x² - 8x) dx, x > 8

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

51. ∫ x²/√(4 + x²) dx

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

33. ∫ √(x² - 9)/x dx, x > 3

Textbook Question

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

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Textbook Question

7–84. Evaluate the following integrals.

33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0

Textbook Question

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).