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
. 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
units thick , then summing up the volumes of all the slices.
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
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
This is also the same as:
Evaluating this intergral,
volume of solid=