Rank–nullity theorem: Difference between revisions

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[[File:Disc integration.svg|thumb|right|170px]]
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{{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.&nbsp;244–248 ({{Google books|Ag26M8TII6oC|online copy|page=244}}{{accessdate|July 12, 2013}})
{{Refend}}
 
 
[[Category:Integral calculus]]
[[Category:Volume]]

Latest revision as of 21:18, 2 October 2014

Over time, the data on your hard drive gets scattered. Defragmenting the difficult drive puts the data into sequential order, making it easier for Windows to access it. As a outcome, the performance of the computer will improve. An good registry cleaner might allow do this task. However in the event you would like to defrag the PC with Windows software. Here a link to show we how.

StreamCI.dll mistakes are caused by a number of different problems, including that the file itself has been moved on a program, the file is outdated or you have installed certain third-party sound motorists which are conflicting with the file. The advantageous news is that if you need to solve the error you're seeing, you need to look to initially confirm the file & drivers are functioning okay on your PC also as then resolving any StreamCI.dll errors which may be inside the registry of your computer.

It doesn't matter whether you're not truly well-defined about what rundll32.exe is. But remember which it plays an significant character in keeping the stability of our computers and the integrity of the program. When certain software or hardware could not answer usually to a system procedure, comes the rundll32 exe error, that could be caused by corrupted files or lost data in registry. Usually, error content might shows up at booting or the beginning of running a program.

Windows errors can be caused by any number of reasons, but there's almost always 1 cause. There's a hidden part of your system that is responsible for creating 90% of all Windows mistakes, and it's called the 'registry'. This really is the central database for a program plus is where the computer shops all its system files and settings. It's a important piece of Windows, which is must be able to function. However, it's moreover one of the largest causes of problems on the PC.

The final step is to create certain that we clean the registry of the computer. The "registry" is a big database that shops important files, settings & choices, and info. Windows reads the files it needs inside order for it to run programs from this database. If the registry gets damaged, infected, or clogged up, then Windows will likely not be capable to correctly access the files it needs for it to load up programs. As this arises, issues plus errors like the d3d9.dll error happen. To fix this and prevent future setbacks, you have to download and run a registry cleaning tool. The highly suggested software is the "Frontline tuneup utilities".

Another element is registry. It is one of the most important piece in a Windows XP, Vista running systems. When Windows commence up, it read connected information from registry and load into computer RAM. This takes up a big part of the startup time. After the data is all loaded, computer runs the business programs.

Perfect Optimizer is a wise Registry Product, updates consistently plus has lots of qualities. Despite its cost, we will find which the update are actually valuable. They provide a lot of help through phone, send plus forums. You might desire to check out the free trial to check it out for yourself.

If you like to have a computer with fast running speed, you'd better install a superior registry cleaner to clean the useless files for you. As long as we take care of your computer, it might keep in good condition.

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