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volume of solid of revolution
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Let us consider a solid of revolution, which is generated when a planar domain  rotates about a line of the same plane. We chose this line for the  -axis, and for simplicity we assume that the boundaries of  are the mentioned axis, two ordinates  ,
 , and a continuous curve  .
Between the bounds  anb  we fit a sequence of points
 and draw through these the ordinates which divide the domain  in  parts. Moreover we form for every part the (maximal) inscribed and the (minimal) circumscribed rectangle. In the revolution of  , each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume  of the union of the cyliders generated by the circumscribed rectangles and at the same time contains the volume  of the union of the cylinders generated by the inscribed rectangles.
Now it is apparent that
where
 are the greatest and
 the least values of the continuous function  on the intervals ![$ [a,\,x_1]$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img20.png "$ [a,\,x_1]$") ,
![$ [x_1,\,x_2]$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img21.png "$ [x_1,\,x_2]$") ,...,
![$ [x_{n-1},\,b]$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img22.png "$ [x_{n-1},\,b]$") . The volume  of the solid of revolution thus satisfies
and this is true for any division
 of the interval ![$ [a,\,b]$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img26.png "$ [a,\,b]$") . The theory of the Riemann integral guarantees that there exists only one real number  having this property and that
it is also the definition of the integral
![$ \displaystyle\int_a^b\!\pi[f(x)]^2\,dx.$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img28.png "$ \displaystyle\int_a^b\!\pi[f(x)]^2\,dx.$") Therefore the volume of the given solid of revolution can be obtained from
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- E. LINDELÖF: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
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"volume of solid of revolution" is owned by pahio.
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(view preamble)
Cross-references: integral, property, real number, Riemann integral, theory, interval, contains, generated by, union, volume, cylinder, circular, rectangle, circumscribed, minimal, inscribed, divide, points, sequence, bounds, curve, continuous, ordinates, axis, boundaries, plane, line, rotates, domain, planar, solid of revolution
There are 2 references to this entry.
This is version 8 of volume of solid of revolution, born on 2007-06-29, modified 2007-06-30.
Object id is 9691, canonical name is VolumeOfSolidOfRevolution.
Accessed 918 times total.
Classification:
| AMS MSC: | 51M25 (Geometry :: Real and complex geometry :: Length, area and volume) |
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Pending Errata and Addenda
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![$\displaystyle V_> = \pi[M_1^2(x_1-a)+M_2^2(x_2-x_1)+\ldots+M_n^2(b-x_{n-1})],$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img15.png "$\displaystyle V_> = \pi[M_1^2(x_1-a)+M_2^2(x_2-x_1)+\ldots+M_n^2(b-x_{n-1})],$"){kind=small}
![$\displaystyle V_< = \pi[m_1^2(x_1-a)+m_2^2(x_2-x_1)+\ldots+m_n^2(b-x_{n-1})],$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img16.png "$\displaystyle V_< = \pi[m_1^2(x_1-a)+m_2^2(x_2-x_1)+\ldots+m_n^2(b-x_{n-1})],$"){kind=small}
![$\displaystyle V = \pi\int_a^b[f(x)]^2\,dx.$](http://images.planetmath.org:8080/cache/objects/9691/l2h/img29.png "$\displaystyle V = \pi\int_a^b[f(x)]^2\,dx.$")