Phong Do
Volumes of Solids Using Common Geometrical Cross Sections
In the disk method, the fundamental element is a disk. The volume is then found by using an integral to sum up all
of the individual volumes of the disks.
The disk method can be extended and adapted to cover the cases where
the object is not a solid of revolution, i.e., the cross section is NOT a circular disk. Your problem may require that
you use
some other common shapes
to define the cross sectional elements such as rectangle, triangle, etc. There
is no specific formula that you can use because each problem is different, depending on the geometry of the cross
section.
The enclosed examples will serve to illustrate the concept.
Below are some representative solids that made up of non-circular cross sections.
A generic guide for problems of this type is summarized below. The area
( )
( )
A x or A y
is specific to a particular
cross section that you need to know. Other than that, the differential
dx or dy
and the limits of integration will
follow the convention used in the disk/washer method.

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