Domanda del libro di testo
90. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / √x, y = 0, x = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (i) about the x-axis
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.2.64b
Consider the region bounded by the graphs of
y = arctan(x), y = 0, and x = 1.
b. Find the volume of the solid formed by revolving this region about the y-axis.
Guida verificata passo dopo passo1
First, identify the region bounded by the curves: \(y = \arctan(x)\), \(y = 0\), and the vertical line \(x = 1\). This region lies between \(x=0\) and \(x=1\) since \(y=0\) corresponds to the \(x\)-axis and \(y=\arctan(x)\) is above it for \(x > 0\).
Since the solid is formed by revolving the region about the \(y\)-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the \(y\)-axis is:
\(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\)
In this problem, the radius of a shell at position \(x\) is the distance from the \(y\)-axis, which is simply \(x\). The height of the shell is the vertical distance between the curves, which is \(\arctan(x) - 0 = \arctan(x)\).
Set up the integral for the volume as:
\(V = \int_0^1 2\pi x \cdot \arctan(x) \, dx\)
To find the volume, you would evaluate the integral \(\int_0^1 x \arctan(x) \, dx\) and then multiply the result by \(2\pi\). Integration by parts is a suitable method to solve this integral.
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Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the given boundaries.
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04:48
Finding Volume Using Disks
Shell Method
The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around a vertical axis. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.
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07:33Euler's Method
Inverse Trigonometric Functions and Their Properties
Understanding the function y = arctan(x) is crucial, including its domain, range, and behavior. Arctan(x) is the inverse of the tangent function, continuous and increasing for all real x, with horizontal asymptotes at ±π/2. This helps in setting integration limits and interpreting the region bounded by the curves.
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06:35Derivatives of Other Inverse Trigonometric Functions
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