CrossSectional Area Method

If
A(x)
is the crosssectional area of a region perpendicular to a
fixed axis at position
x
, defined between
x = a
and
x = b
, then the
total volume of the region is
Vol =
A(x)dx


Disk Method

The volume of the region obtained by rotating the area under the graph
of a function
f (x)
between
x = a
and
x = b
about the
x
axis is
equal to
Vol = 2Π
f (x)^{2}
dx


This is an application of the crosssectional area method, noting
that that cross section to this surface of revolution perpendicular to
the
x
axis is a circle of radius
f (x)
.
Shell Method

The volume of the region obtained by rotating the area under a
function
f (x)
between
x = a
and
x = b
about the
y
axis is equal
to:
Vol = 2Π
xf (x)dx


Solid of Revolution

The solid swept out by a region in the plane when rotated about an
axis. Examples include cylinders, cones, and spheres (all considered
as solids with their interiors). The volume of such a region can be
computed via the disk method or the shell method.
Surface of Revolution

The surface swept out by a curve in the plane when rotated about an
axis. Examples include the surface of a cylinder, cone, or sphere,
and more generally the surface of any solid of revolution.