# 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: