![Disc integration](https://upload.wikimedia.org/wikipedia/commons/a/a8/Disc_integration.svg)
Disc integration, also known in
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
as the disc method, is a method for calculating the
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of a
solid of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...
of a solid-state material when
integrating along an axis "parallel" to the
axis of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...
. 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
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis ''perpendicular to'' the axis of revolution. This is in contrast to disc integration whic ...
, which integrates along an axis ''perpendicular'' to the axis of revolution.
Definition
Function of
If the function to be revolved is a function of , the following integral represents the volume of the solid of revolution:
:
where is the distance between the function and the axis of rotation. This works only if the
axis of rotation
Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
is horizontal (example: or some other constant).
Function of
If the function to be revolved is a function of , the following integral will obtain the volume of the solid of revolution:
:
where is the distance between the function and the axis of rotation. This works only if the
axis of rotation
Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
is vertical (example: 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 is the function that is farthest from the axis of rotation and is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the -axis of the red "leaf" enclosed between the square-root and quadratic curves:
![Solid of revolution](https://upload.wikimedia.org/wikipedia/commons/c/c6/Solid_of_revolution.gif)
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 -axis.)
To rotate about any horizontal axis, simply subtract from that axis from each formula. If is the value of a horizontal axis, then the volume equals
:
For example, to rotate the region between and along the axis , 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 , 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 is, with respect to the x-axis, further up than the graph of , with respect to the axis of rotation the function is the inner function: its graph is closer to or the equation of the axis of rotation in the example.
The same idea can be applied to both the -axis and any other vertical axis. One simply must solve each equation for before one inserts them into the integration formula.
See also
*
Solid of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...
*
Shell integration
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis ''perpendicular to'' the axis of revolution. This is in contrast to disc integration whic ...
References
*
*
*
Frank Ayres,
Elliott Mendelson
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard Unive ...
. ''
Schaum's Outlines
Schaum's Outlines () is a series of supplementary texts for American high school, AP, and college-level courses, currently published by McGraw-Hill Education Professional, a subsidiary of McGraw-Hill Education. The outlines cover a wide variety of ...
: Calculus''. McGraw-Hill Professional 2008, . pp. 244–248 (. Retrieved July 12, 2013.)
{{Calculus topics
Integral calculus
Volume