World Library  
Flag as Inappropriate
Email this Article

Disc integration

Article Id: WHEBN0000330603
Reproduction Date:

Title: Disc integration  
Author: World Heritage Encyclopedia
Language: English
Subject: Mean value theorem, Calculus, Volume, Integral calculus, Integral
Collection: Integral Calculus, Volume
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Disc integration

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

Contents

  • Definition 1
    • Function of x 1.1
    • Function of y 1.2
    • Washer method 1.3
  • See also 2
  • References 3

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:

\pi\int_a^b [R(x)]^2\ \mathrm{d}x

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:

\pi\int_c^d [R(y)]^2\ \mathrm{d}y

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:

\pi\int_a^b(\left[R_O(x)\right]^2 - \left[R_I(x)\right]^2)\mathrm{d}x

where R_O(x) is the function that is farthest from the axis of rotation and R_I(x) is the function that is closest to the axis of rotation. 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.

[R_O(x)]^2 - [R_I(x)]^2\ \not\equiv\; [R_O(x) - R_I(x)]^2

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

\pi\int_a^b([h-R_O(x)]^2 - [h-R_I(x)]^2)\,\mathrm{d}x.

For example, to rotate the region between y=-2x+x^2 and y=x

along the axis y=4, one would integrate as follows:

\pi\int_0^3([4-(-2x+x^2)]^2 - [4-x]^2)\,\mathrm{d}x.

The bounds of integration are the zeros of the first equation minus the second. Note that when you integrate along an axis other than the x, the further axis may not be that obvious. In the previous example, even though y=x is further up than y=-2x+x^2, it is the inner axis since it is closer to y=4

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

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.