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 8.7.12a
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
Verified step by step guidance1
Identify the function to be integrated: \(f(x) = 2x - 1\) over the interval \([1, 3]\).
Recall the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
Calculate the second derivative of the function: \(f''(x) = \frac{d^2}{dx^2}(2x - 1)\). Since \(f(x)\) is linear, \(f''(x) = 0\) for all \(x\).
Since \(f''(x) = 0\), the error bound formula implies the Trapezoidal Rule will give the exact value regardless of \(n\), so the minimum number of subintervals needed to achieve an error less than \(10^{-4}\) is 1.
Conclude that for this linear function, the Trapezoidal Rule is exact with just one subinterval, so no further subdivision is necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trapezoidal Rule
The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.
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Power Rules
Error Bound for the Trapezoidal Rule
The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. It is given by |E| ≤ (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps estimate how many subintervals are needed to achieve a desired accuracy.
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04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the integrand measures the function's concavity and affects the error in the Trapezoidal Rule. A smaller maximum second derivative on the interval means less curvature and thus a smaller error, allowing fewer subintervals for a given accuracy.
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06:02The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
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?
1
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Textbook Question
Consider the region bounded by the graphs of
y = ln(x), y = 0, and x = e.
a. Find the area of the region.
1
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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
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.