Volume of Revolution
From Math Images
- This image is a solid of revolution
Solid of revolution |
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Contents |
Basic Description
This image shows the solid formed after revolving the region bounded by , , and , about the -axisA 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, [...]
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.
In the disc method, we imagine chopping up the solid into thin cylindrical plates, each
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 , , and
<-------Plotting the graph of this area,
To find the volume of the solid using the disc method:
Volume of one disc = where - which is the function- is the radius of the circular cross-section and is the thickness of each disc
Volume of all dics:
Volume of all discs = , with ranging from 0 to 1
If we make the slices infinitesmally thick, the Riemann sum becomes the same as:
Evaluating this intergral,
volume of solid=
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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