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 
( 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:


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