# Geometrical and mechanical applications of definite integral

*Volume of revolution body. Work of variable force.*

A definite integral has numerous applications in mathematics, mechanics, physics, astronomy, engineering and other fields of human activities. We’ll consider here only two examples, illustrating possibilities of this apparatus.

*
A volume of revolution body.
*
Consider a body, received by revolution around an axis
*
OX
*
of a curvilinear trapezoid, bounded by the
*
*
function
*
f
*
(
*
x
*
) graph,straight lines
*
x
*
=
*
a
*
and
*
x
*
=
*
b
*
and an axis
*
OX
*
( Fig.10).

A volume
*
V
*
of the revolution body is:

*
A work of variable force.
*
Consider motion of a material point, moving along an axis
*
OX
*
under the influence of variable force
*
f
*
, depending on
the point position
*
x
*
on the axis, i.e. the force is a function of
*
x
*
. Then a work
*
A
*
, necessary to move the material point from the position
*
x
*
=
*
a
*
to the position
*
x
*
=
*
b
*
is calculated by the formula:

E x a m p l e . Find a volume of a truncated cone, formed by revolution of the straight line

*
y
*
=
*
x
*
+ 1 around an axis
*
OX
*
and bounded by lines
*
x
*
= 0 and
*
x
*
= 3 .

S o l u t i o n . According to the above represented formula we have: