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: