Integral with variable upper limit of integration
Let f ( x) be a continuous function, given in a
segment [ a , b ], then for any x
[ a , b ] the function
exists. This function is given as an integral with variable upper limit of integration
in the right-hand part of the equality.
All rules and properties of a definite integral apply to an integral with variable upper limit of integration.
|E x a m p l e .||A variable force acting on a linear way changes in the law:
x ) = 6x2 + 5 at x
0 . What law does a work of this force change in ?
|S o l u t i o n.||A work of the force f ( x ) on a segment [ 0 , x ] of linear way is equal to:
Thus, the work changes in
the law: F ( x) = 2x 3 + 5x.
According to the definition of an integral with variable upper limit of integration or the
function F ( x ) and known properties of an integral it follows that at x
[ a , b]
F’ ( x ) = f ( x ) .
Check this property using the above mentioned example.