Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
Ch. 7 - Transcendental Functions
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 7, Problem 7.AAE.3
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to 0^+} \left( \cos(\sqrt{x}) \right)^{\frac{1}{x}}\).
Recognize that the expression is of the form \(f(x)^{g(x)}\) where both the base and the exponent depend on \(x\), and direct substitution leads to an indeterminate form.
Rewrite the limit using the exponential and natural logarithm to handle the exponent: \(\lim_{x \to 0^+} e^{\frac{1}{x} \ln(\cos(\sqrt{x}))}\).
Focus on finding the limit of the exponent: \(\lim_{x \to 0^+} \frac{\ln(\cos(\sqrt{x}))}{x}\). Use series expansions or approximations for \(\cos(\sqrt{x})\) near 0 to simplify the logarithm.
Apply the appropriate limit techniques (such as L'Hôpital's Rule if needed) to evaluate the exponent limit, then substitute back into the exponential to find the original limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a function as x approaches a point
The limit describes the value that a function approaches as the input approaches a specific point. Understanding one-sided limits, such as x approaching 0 from the right (0⁺), is crucial when the function behaves differently on either side of the point.
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Limits of Rational Functions: Denominator = 0
Behavior of composite functions and substitution
When dealing with limits involving composite functions like cos(√x), it helps to analyze the inner function's behavior first. Substituting variables or expressions can simplify the limit evaluation, especially when the argument approaches zero.
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3:48Evaluate Composite Functions - Special Cases
Exponential limits and the form 1^∞
Limits of the form (f(x))^(g(x)) where the base approaches 1 and the exponent approaches infinity often lead to an indeterminate form 1^∞. Applying logarithms and using L'Hôpital's Rule or series expansions helps to resolve such limits.
Recommended video:
6:13Exponential Functions
Related Practice
Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?
Textbook Question
19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.
Textbook Question
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
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Textbook Question
Find the limits in Exercises 1–6.
1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)
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