Textbook Question
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.9.31
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv
Verified step by step guidance1
Identify that the integral \( \int_1^{\infty} \frac{1}{v(v+1)} \, dv \) is an improper integral because the upper limit is infinite.
Rewrite the integral as a limit: \( \lim_{t \to \infty} \int_1^{t} \frac{1}{v(v+1)} \, dv \). This allows us to handle the infinite limit properly.
Use partial fraction decomposition to simplify the integrand: express \( \frac{1}{v(v+1)} \) as \( \frac{A}{v} + \frac{B}{v+1} \) and solve for constants \( A \) and \( B \).
Integrate the decomposed expression term-by-term over \( v \) from 1 to \( t \), resulting in logarithmic functions.
Evaluate the definite integral by substituting the limits 1 and \( t \), then take the limit as \( t \to \infty \) to determine if the integral converges or diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence depends on whether this limit exists finitely.
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Improper Integrals: Infinite Intervals
Partial Fraction Decomposition
Partial fraction decomposition breaks a complex rational function into simpler fractions that are easier to integrate. For example, 1/[v(v + 1)] can be rewritten as A/v + B/(v + 1), allowing straightforward integration of each term separately.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Evaluating Limits of Integrals
When dealing with improper integrals, after integrating the function, you must evaluate the limit of the antiderivative as the variable approaches infinity. This step determines if the integral converges to a finite value or diverges.
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05:50One-Sided Limits
Related Practice
Textbook Question
23-64. Integration Evaluate the following integrals.
26. ∫₀¹ [1 / (t² - 9)] dt
Textbook Question
7–64. Integration review Evaluate the following integrals.
47. ∫ dx / (x⁻¹ + 1)
Textbook Question
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)
Textbook Question
7–64. Integration review Evaluate the following integrals.
62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx
Textbook Question
69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?
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