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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.4.51b
Capitolo 6, Problema 6.4.51b

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
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b. Use the washer method to write an integral for the volume of the torus.

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Identify the region being revolved: The circle of radius 2 is centered at (3, 0) in the xy-plane. Its equation is \((x - 3)^2 + y^2 = 2^2 = 4\).
Since the circle is revolved about the y-axis, we use the washer method with respect to the y-axis. The washers will be horizontal slices perpendicular to the y-axis.
Express the x-values (radii) in terms of y. From the circle equation, solve for \(x\): \(x = 3 \pm \sqrt{4 - y^2}\). The outer radius \(R(y)\) is the distance from the y-axis to the outer edge of the circle, which is \(3 + \sqrt{4 - y^2}\), and the inner radius \(r(y)\) is \(3 - \sqrt{4 - y^2}\).
The volume element for the washer method when revolving around the y-axis is \(\pi (R(y)^2 - r(y)^2) \, dy\). Determine the limits of integration for \(y\), which are the vertical bounds of the circle: from \(-2\) to \(2\).
Write the integral for the volume as \(V = \int_{-2}^{2} \pi \left[ (3 + \sqrt{4 - y^2})^2 - (3 - \sqrt{4 - y^2})^2 \right] \, dy\).

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Washer Method for Volume

The washer method calculates the volume of a solid of revolution by integrating the difference between the outer and inner radii squared, multiplied by π, along the axis of rotation. It is used when the solid has a hole, forming washers instead of disks. The volume integral sums these washers' volumes across the interval.
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Finding Volume Using Disks

Setting up the Integral with Respect to y-axis

When revolving a region around the y-axis, the integral is typically set up in terms of y. The radius of each washer is determined by the x-values expressed as functions of y. Understanding how to express the boundaries of the region in terms of y is essential for correctly defining the limits and radii in the integral.
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Disk Method Using y-Axis

Equation of the Circle and Its Position

The circle with radius 2 centered at (3,0) is described by (x-3)^2 + y^2 = 4. This equation helps find the outer and inner radii of the washers when revolving around the y-axis. Recognizing the circle's position relative to the axis of rotation is crucial for determining the correct radii in the volume integral.
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Parameterizing Equations of Circles & Ellipses
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Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


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