Textbook Question
17. Even-odd decompositions
b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that
f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2
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 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 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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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
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