Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.
a. About the y-axis.
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.9.42a
Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.
a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?
Verified step by step guidance1
Identify the given information: the lifetime L of tires is normally distributed with mean \( \mu = 22000 \) miles and standard deviation \( \sigma = 4000 \) miles. The batch size is 4000 tires.
We want to find the number of tires expected to last at least 18,000 miles. This means we need to find \( P(L \geq 18000) \), the probability that a tire lasts 18,000 miles or more.
Convert the lifetime value 18,000 miles to a standard normal variable \( Z \) using the formula: \[ Z = \frac{X - \mu}{\sigma} = \frac{18000 - 22000}{4000} \]. This standardizes the value to the standard normal distribution.
Use the standard normal distribution table or a calculator to find \( P(Z \geq z) \), where \( z \) is the value calculated in the previous step. This gives the probability that a tire lasts at least 18,000 miles.
Multiply the probability \( P(L \geq 18000) \) by the total number of tires (4000) to find the expected number of tires lasting at least 18,000 miles: \[ \text{Expected number} = 4000 \times P(L \geq 18000) \].
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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 its bell-shaped curve, defined by the mean (μ) and standard deviation (σ). It models many natural phenomena, including tire lifetimes, and allows calculation of probabilities for values within certain ranges.
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Standardization (Z-Score)
Standardization converts a normal random variable to a standard normal variable (Z) with mean 0 and standard deviation 1. This is done using the formula Z = (X - μ) / σ, enabling the use of standard normal tables to find probabilities associated with specific values.
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Expected Value in Probability
The expected value represents the average or mean outcome in a probabilistic scenario. When dealing with large batches, multiplying the probability of an event by the total number of trials estimates how many items meet a certain criterion, such as tires lasting at least 18,000 miles.
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Related Practice
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 3 of (2x - 1) dx
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of (t³ + t) dt
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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 1) dx
Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about
a. The y-axis.
(See Exercise 57 for a graph.)
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.