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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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 8.8.81a
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
Verified step by step guidance1
Identify the integral and the parameter involved: we have the integral \( \int_1^2 \frac{dx}{x (\ln x)^p} \), and we want to find for which values of \( p \) this integral converges.
Check the behavior of the integrand near the limits of integration. Since the interval is from 1 to 2, and \( \ln 1 = 0 \), the potential issue is at the lower limit \( x = 1 \) where the denominator involves \( (\ln x)^p \) which tends to zero, possibly causing a singularity.
Make a substitution to analyze the behavior near \( x = 1 \): let \( t = \ln x \). Then \( dt = \frac{1}{x} dx \), so \( dx = x dt = e^t dt \). The integral limits change from \( x=1 \) to \( t=0 \), and from \( x=2 \) to \( t=\ln 2 \).
Rewrite the integral in terms of \( t \): \( \int_0^{\ln 2} \frac{e^t dt}{e^t t^p} = \int_0^{\ln 2} \frac{dt}{t^p} \). Now the integral simplifies to \( \int_0^{\ln 2} t^{-p} dt \).
Determine convergence of \( \int_0^{a} t^{-p} dt \) for some positive \( a \). This integral converges if and only if \( -p > -1 \), or equivalently \( p < 1 \). So the original integral converges for \( p < 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals and Convergence
An improper integral involves limits where the integrand becomes unbounded or the interval is infinite. To determine convergence, we analyze the behavior of the integrand near points causing potential issues, such as where the denominator approaches zero or infinity.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Behavior of the Natural Logarithm Function near 1
The natural logarithm ln(x) approaches 0 as x approaches 1 from the right. Since the integral's denominator includes (ln x)^p, understanding how ln(x) behaves near 1 is crucial to assess whether the integral converges or diverges.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
p-Test for Convergence of Integrals
The p-test helps determine convergence of integrals of the form ∫ (1 / t^p) dt near zero or infinity. By substituting t = ln(x), the integral can be analyzed using the p-test to find values of p for which the integral converges.
Recommended video:
07:51Choosing a Convergence Test
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 1 of (t³ + 1) 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 1 to 2 of 1/s² ds
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.
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 -1 to 1 of (x² + 1) dx
Textbook Question
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.