Volume of Revolution

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|Image=VolumeOfRev.jpg
|Image=VolumeOfRev.jpg
|ImageIntro=This image shows a solid of revolution
|ImageIntro=This image shows a solid of revolution
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|ImageDescElem=This image shows the solid formed after revolving the region bounded by <math>y=x^2</math>, <math>y=0</math>,<math>x=0</math> and <math>x=1</math>
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|ImageDescElem=This image shows the solid formed after revolving the region bounded by <math>y=x^2</math>, <math>y=0</math>,<math>x=0</math> and <math>x=1</math>, about the <math>x</math>-axis
|ImageDesc=When finding the volume of revolution of solids, in many cases the problem is not with the calculus, but with actually visualizing the solid. To find the volume of a solid like a cylinder, usually we use the formula <math>{\pi} {r^2} h</math>. Alternatively we can imagine chopping up the cylinder into thin cylindrical plates, much like slicing up bread, computing the volume of each slice, each of which is <math>{\Delta x }</math> units thick , then summing up the volumes of all the slices.[[Image:bread.gif|left|thumb| The disc method is much like slicing up bread and computing the volume of each slice http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html]]
|ImageDesc=When finding the volume of revolution of solids, in many cases the problem is not with the calculus, but with actually visualizing the solid. To find the volume of a solid like a cylinder, usually we use the formula <math>{\pi} {r^2} h</math>. Alternatively we can imagine chopping up the cylinder into thin cylindrical plates, much like slicing up bread, computing the volume of each slice, each of which is <math>{\Delta x }</math> units thick , then summing up the volumes of all the slices.[[Image:bread.gif|left|thumb| The disc method is much like slicing up bread and computing the volume of each slice http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html]]
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Volume of all discs = <math>{\sum}{\pi}y^2{\Delta x}</math>, with <math>X</math> ranging from 0 to 1
Volume of all discs = <math>{\sum}{\pi}y^2{\Delta x}</math>, with <math>X</math> ranging from 0 to 1
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This is also the same as:
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If we make the slices infinitesmally thick, the Riemann sum becomes the same as:
<math>\int_0^1 {\pi}y^2\,dx ={\pi}\int_0^1 (x^2)^2\, dx</math>
<math>\int_0^1 {\pi}y^2\,dx ={\pi}\int_0^1 (x^2)^2\, dx</math>

Revision as of 13:57, 8 July 2009


Solid of revolution
This image shows a solid of revolution

Contents

Basic Description

This image shows the solid formed after revolving the region bounded by y=x^2, y=0,x=0 and x=1, about the x-axis

A More Mathematical Explanation

Note: understanding of this explanation requires: *Pre-calculus and elementary calculus

When finding the volume of revolution of solids, in many cases the problem is not with the calculus, [...]

When finding the volume of revolution of solids, in many cases the problem is not with the calculus, but with actually visualizing the solid. To find the volume of a solid like a cylinder, usually we use the formula {\pi} {r^2} h. Alternatively we can imagine chopping up the cylinder into thin cylindrical plates, much like slicing up bread, computing the volume of each slice, each of which is {\Delta x } units thick , then summing up the volumes of all the slices.
The disc method is much like slicing up bread and computing the volume of each slice http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html
The disc method is much like slicing up bread and computing the volume of each slice http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html

This is the idea behind computing the volume of solids whose shapes are complicated. If we are given a function which decribes the shape of a solid, we can plot the function, then revolve the resultant plane area about a fixed axis to obtain the original solid, now called the the solid of revolution. The area can be bounded by many curves of almost any form. The volume of the solid can then be computed using the disc method.

Note: There are other ways of computing the volumes of complicated solids other than the disc method.

In the disc method, we imagine chopping up the solid into thin cylindrical plates, each

{\Delta x } units thick, calculating the volume of each plate, then summing up the volumes of all plates.

For example, let's consider a region bounded by y=x^2, y=0,x=0 and x=1


<-------Plotting the graph of this area,


If we revolve this area about the x axis (y=0), then we get the main image on the right hand side of the page
This image shows a  plane area being revolved to create a solid http://curvebank.calstatela.edu/volrev/volrev.htm
This image shows a plane area being revolved to create a solid http://curvebank.calstatela.edu/volrev/volrev.htm

To find the volume of the solid using the disc method:

Volume of one disc = {\pi} y^2{\Delta x} where y- which is the function- is the radius of the circular cross-section and \Delta x is the thickness of each disc

Volume of all dics:


Volume of all discs = {\sum}{\pi}y^2{\Delta x}, with X ranging from 0 to 1

If we make the slices infinitesmally thick, the Riemann sum becomes the same as:

\int_0^1 {\pi}y^2\,dx ={\pi}\int_0^1 (x^2)^2\, dx

Evaluating this intergral,


{\pi}\int_0^1 x^4 dx

=[{{x^5\over 5} + C|}_0^1] {\pi}

=[{1\over 5} + {0\over 5}] {\pi}

={\pi}\over 5

volume of solid= {\pi\over 5} units^3


References

Bread image http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html
Revolving image http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html




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