Applications of the Integral
Terms
Cross-Sectional Area Method
-
If
A(x)
is the cross-sectional 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
This is an application of the cross-sectional area method, noting that that cross section to this surface of revolution perpendicular to the x -axis is a circle of radius f (x) .
|
Vol = 2Π
f (x)2
dx
|
This is an application of the cross-sectional 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.
