Volume of Revolution
From Math Images
|Solid of revolution|
Basic DescriptionThis image shows the solid formed after revolving the region bounded by , , and , about the -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, [...]
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.
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,
If we revolve this area about the x axis (), then we get the main image on the right hand side of the page
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=
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