Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.9.40
Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?
Verified step by step guidance1
Identify the random variable \( X \) representing the annual rainfall, which follows a normal distribution with mean \( \mu = 20.11 \) inches and standard deviation \( \sigma = 4.7 \) inches.
Standardize the value 17 inches to find the corresponding z-score using the formula:
\[ Z = \frac{X - \mu}{\sigma} = \frac{17 - 20.11}{4.7} \]
Calculate the z-score to determine how many standard deviations 17 inches is below the mean rainfall.
Use the standard normal distribution table (or a calculator) to find the probability \( P(Z \leq z) \) corresponding to the calculated z-score.
Since the problem asks for the probability that rainfall exceeds 17 inches, compute \( P(X > 17) = 1 - P(Z \leq z) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve defined by its mean and standard deviation. It models many natural phenomena, including rainfall amounts, allowing us to calculate probabilities for ranges of values.
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Standardization (Z-Score)
Standardization converts a normal random variable to a standard normal variable with mean 0 and standard deviation 1 using the formula Z = (X - μ) / σ. This transformation enables the use of standard normal tables or software to find probabilities.
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Probability Calculation for Normal Variables
To find the probability that a normal variable exceeds a certain value, we calculate the corresponding Z-score and then use the standard normal distribution to find the area to the right of that Z. This area represents the probability of exceeding the given value.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / ((x¹/³ - 1)√x) dx
(Hint: Let x = u⁶.)
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx
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Textbook Question
What is the largest value that
∫ from a to b x√(2x - x²) dx
can have for any a and b? Give reasons for your answer.
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