Textbook Question
Moment about y-axis:
A thin plate of constant density δ = 1 occupies the region enclosed by the curve
y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.
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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 8.8.58
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 2 to ∞ of ((x dx) / √(x⁴ - 1))
Verified step by step guidance1
First, write down the integral clearly: \(\int_{2}^{\infty} \frac{x}{\sqrt{x^{4} - 1}} \, dx\).
To determine convergence, analyze the behavior of the integrand as \(x \to \infty\). Simplify the expression inside the square root for large \(x\): \(\sqrt{x^{4} - 1} \approx \sqrt{x^{4}} = x^{2}\).
Using this approximation, the integrand behaves like \(\frac{x}{x^{2}} = \frac{1}{x}\) for large \(x\). This suggests comparing the integral to \(\int_{2}^{\infty} \frac{1}{x} \, dx\).
Apply the Direct Comparison Test or Limit Comparison Test with the function \(g(x) = \frac{1}{x}\). Since \(\int_{2}^{\infty} \frac{1}{x} \, dx\) diverges (harmonic integral), use the Limit Comparison Test to confirm the behavior of the original integral.
Compute the limit \(\lim_{x \to \infty} \frac{\frac{x}{\sqrt{x^{4} - 1}}}{\frac{1}{x}} = \lim_{x \to \infty} \frac{x^{2}}{\sqrt{x^{4} - 1}}\). Simplify this limit to determine if it is a finite positive number, which will indicate that both integrals share the same convergence behavior.
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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 convergence, we consider the limit of the integral as the upper bound approaches infinity. Determining whether the integral converges or diverges is essential for understanding the behavior of functions over unbounded domains.
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Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test determines convergence by comparing the given integral to a simpler integral with known behavior. If the integrand is less than or equal to a convergent function, the integral converges; if it is greater than or equal to a divergent function, it diverges. This test requires identifying appropriate bounding functions.
Recommended video:
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test compares two functions by examining the limit of their ratio as the variable approaches infinity. If the limit is a positive finite number, both integrals either converge or diverge together. This test is useful when direct comparison is difficult but the asymptotic behavior of the integrand is known.
Recommended video:
07:45Limit Comparison Test
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
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.
∫ (dx / ((x - 2)√(x² - 4x + 3)))
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Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (4 + x²)
Textbook Question
87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)