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Rank–nullity theorem

2014-01-29T11:41:48Z

137.205.238.248:

[[File:Disc integration.svg|thumb|right|170px]]
{{Calculus |Integral}}

'''Disk integration''', also known in [[integral calculus]] as the '''disk method''', is a means of calculating the [[volume]] of a [[solid of revolution]] of a solid-state material when [[Integral|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.

== Definition ==

===Function of ''x''===
If the function to be revolved is a function of <math>x</math>, the following integral represents the volume of the solid of revolution:

: <math>\pi \int_a^b [R(x)]^2\ \mathrm{d}x</math>

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

===Function of ''y''===
If the function to be revolved is a function of <math>y</math>, the following integral will obtain the volume of the solid of revolution:

: <math>\pi \int_c^d [R(y)]^2\ \mathrm{d}y</math>

where <math>R(y)</math> is the distance between the function and the axis of rotation. This works only if the [[axis of rotation]] is vertical (example: <math>x=4</math> 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:

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

where <math>R_O(x)</math> is the function that is farthest from the axis of rotation and <math>R_I(x)</math> 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.

: <math> [R_O(x)]^2 - [R_I(x)]^2\ \not\equiv \; [R_O(x) - R_I(x)]^2</math>

''The above formula only works for revolutions about the x-axis.''

To rotate about any horizontal axis, simply subtract from that axis each formula:

if <math>h</math> is the value of a horizontal axis, then the volume =

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

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

along the axis <math>y=4</math>, one would integrate as follows:

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

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 <math>x</math>, the further axis may not be that obvious. In the previous example, even though <math>y=x</math> is further up than <math> y=-2x+x^2 </math>, it is the inner axis since it is closer to <math>y=4</math>

The same idea can be applied to both the y-axis and any other vertical axis. You simply must solve each equation for <math>x</math> before you insert them into the integration formula.

== See also ==
* [[Solid of revolution]]
* [[Shell integration]]

==References==
{{Refbegin|normalfont=yes}}
*{{cite web|title=Volumes of Solids of Revolution|url=http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html|work=CliffsNotes.com|accessdate=July 12, 2013}}
*{{MathWorld|title=Method of Disks|urlname=MethodofDisks}} {{accessdate|July 12, 2013}}
*Frank Ayres, Elliott Mendelson. ''Schaum's outlines: Calculus''. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 ({{Google books|Ag26M8TII6oC|online copy|page=244}}{{accessdate|July 12, 2013}})
{{Refend}}

[[Category:Integral calculus]]
[[Category:Volume]]

Nth root

2013-12-12T14:34:08Z

137.205.238.132:

{{Signs of death}}

'''Algor mortis''' ([[Latin]]: ''algor''—coldness; ''mortis''—of death) is the reduction in body temperature following [[death]]. This is generally a steady decline until matching [[ambient temperature]], although external factors can have a significant influence.

== Use in forensic science and crime solving ==

A measured rectal temperature can give some indication of the time of death. Although the [[heat conduction]] which leads to body cooling follows an [[exponential decay]] curve, it can be approximated as a linear process: 2° Celsius during the first hour and 1° Celsius per hour until the body nears ambient temperature.

The '''Glaister equation'''<ref>http://www.fmap.archives.gla.ac.uk/DC403/DC403_page.htm</ref><ref>{{cite book|title=Forensic Medicine|author=Guharaj, P. V.|chapter=Cooling of the body (algor mortis)|year=2003|edition=2nd|pages=61–62|publisher=Longman Orient|location=Hyderabad|url=http://books.google.com/books?isbn=8125024883}}</ref> estimates the hours elapsed since death as a [[linear function]] of the [[rectal temperature]]:

:<math>
\frac{98.4\,^{\circ}{\rm F} - \text{rectal temperature in Fahrenheit}}{1.5}
</math>


As [[decomposition]] occurs the internal body temperature tends to rise again.

== Factors affecting the rate of algor mortis ==
Generally, temperature change is considered an inaccurate means of determining time of death, as the rate of change is affected by several key factors including:<ref>http://www.crimewatchcanada.com/index.php?option=com_content&view=article&id=79:changes-to-a-body-after-death&catid=41:january-february-2010-issue-76&Itemid=53</ref>
* Stability or fluctuation of the ambient temperature.
* The thermal conductivity of the surface on which a body lies.
* Diseases or drugs which increase body temperature raising the starting temperature of the corpse at the time of death
* The existence of a "temperature plateau",<ref>{{cite journal|last=Kaliszan|first=M.|title=Verification of the exponential model of body temperature decrease after death in pigs|journal=Experimental Physiology|date=20 May 2005|volume=90|issue=5|pages=727–738|doi=10.1113}}</ref> a highly variable period of time in which the body does not cool.

==References==
{{reflist}}
* {{cite book | first=Richard| last=Saferstein| year=2004| title=Criminalistics An Introduction to Forensic Science| edition=8th| publisher=Pearson Prentice Hall| isbn=0-13-113706-9 }}
* Karen T. Taylor, "Forensic art and illustration", CRC Press, 2000, ISBN 0-8493-8118-5, p. 308
* Robert G. Mayer, "Embalming: history, theory, and practice", McGraw-Hill Professional, 2005, ISBN 0-07-143950-1, p. 106
* Calixto Machado, "Brain death: a reappraisal", Springer, 2007, ISBN 0-387-38975-X, pp. 73–74

==External links==
* [http://www.arrakis.es/~jacoello/date.pdf Standards Employed to Determine Time of Death]

{{DEFAULTSORT:Algor Mortis}}
[[Category:Signs of death]]
[[Category:Latin medical phrases]]

{{med-sign-stub}}
{{death-stub}}
{{Latin-vocab-stub}}