To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Disc integration

From Wikipedia, the free encyclopedia

Disc integration.svg

Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions. This is in contrast to shell integration, which integrates along an axis perpendicular to the axis of revolution.

Definition

Function of x

If the function to be revolved is a function of x, the following integral represents the volume of the solid of revolution:

where R(x) is the distance between the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: y = 3 or some other constant).

Function of y

If the function to be revolved is a function of y, the following integral will obtain the volume of the solid of revolution:

where R(y) is the distance between the function and the axis of rotation. This works only if the axis of rotation is vertical (example: x = 4 or some other constant).

Washer method

To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:

where RO(x) is the function that is farthest from the axis of rotation and RI(x) is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the x-axis of the red "leaf" enclosed between the square-root and quadratic curves:

Rotation about x-axis
Rotation about x-axis

The volume of this solid is:

One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

(This formula only works for revolutions about the x-axis.)

To rotate about any horizontal axis, simply subtract from that axis each formula. If h is the value of a horizontal axis, then the volume equals

For example, to rotate the region between y = −2x + x2 and y = x along the axis y = 4, one would integrate as follows:

The bounds of integration are the zeros of the first equation minus the second. Note that when integrating along an axis other than the x, the graph of the function that is farthest from the axis of rotation may not be that obvious. In the previous example, even though the graph of y = x is, with respect to the x-axis, further up than the graph of y = −2x + x2, with respect to the axis of rotation the function y = x is the inner function: its graph is closer to y = 4 or the equation of the axis of rotation in the example.

The same idea can be applied to both the y-axis and any other vertical axis. One simply must solve each equation for x before one inserts them into the integration formula.

See also

References

  • "Volumes of Solids of Revolution". CliffsNotes.com. Retrieved July 8, 2014.
  • Weisstein, Eric W. "Method of Disks". MathWorld.
  • Frank Ayres, Elliott Mendelson. Schaum's Outlines: Calculus. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 (online copy, p. 244, at Google Books. Retrieved July 12, 2013.)
  • "The Disk and Washer Methods" Avidemia.com
This page was last edited on 16 January 2021, at 09:57
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.