Volume of Revolution

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|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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This is the idea behind computing the volume of solids whose shapes are complicated. Given a function, we can graph it, then revolve the plane area we get about a fixed axis to obtain a solid called the '''solid of revolution'''.In general, the plane area can be bounded by many curves of almost any form. The volume of the solid can then be computed using the '''disc method'''. <br>
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In general, given a function, we can graph it then revolve the area under the curve between two specific x-coordinates about a fixed axis to obtain a solid called the '''solid of revolution'''. The volume of the solid can then be computed using the '''disc method'''. <br>
Note: There are other ways of computing the volumes of complicated solids other than the disc method.<br> [[Image:area.jpg|left|thumb|200px]]
Note: There are other ways of computing the volumes of complicated solids other than the disc method.<br> [[Image:area.jpg|left|thumb|200px]]

Revision as of 15:40, 8 July 2009


Solid of revolution
This image is a solid of revolution
Solid of revolution
Field: Calculus
Created By: Lizah Masis

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

In general, given a function, we can graph it then revolve the area under the curve between two specific x-coordinates about a fixed axis to obtain a solid called the solid of revolution. 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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