Textbook Question
Find the limits in Exercises 1–6.
3. lim(x→0⁺) (cox(√x))^(1/x)
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.20a
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.
Verified step by step guidance1
Identify the function and the interval: The curve is given by \(y = \frac{1}{2\sqrt{x}}\) and the interval is from \(x = \frac{1}{4}\) to \(x = 4\).
Recall the formula for the volume of a solid of revolution about the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the function being revolved.
Square the function inside the integral: \(\left( \frac{1}{2\sqrt{x}} \right)^{2} = \frac{1}{4x}\).
Set up the integral for the volume: \(V = \pi \int_{\frac{1}{4}}^{4} \frac{1}{4x} \, dx\).
Evaluate the integral by integrating \(\frac{1}{4x}\) with respect to \(x\) over the interval \(\left[ \frac{1}{4}, 4 \right]\), then multiply the result by \(\pi\) to find the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solid of Revolution
A solid of revolution is formed when a plane region is rotated about a line (axis), creating a three-dimensional object. The volume of such solids can be found using integral calculus by summing infinitesimal disks or washers perpendicular to the axis of rotation.
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Finding Volume Using Disks
Disk Method
The disk method calculates the volume of a solid of revolution by slicing it into thin circular disks perpendicular to the axis. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.
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06:30Disk Method Using y-Axis
Definite Integration
Definite integration computes the exact accumulation of quantities, such as area or volume, over a specific interval. In this problem, integrating the squared function from x=1/4 to x=4 calculates the total volume generated by revolving the region around the x-axis.
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05:43Definition of the Definite Integral
Related Practice
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
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)
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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