Integral with variable upper limit of integration
) be a continuous function, given in a
], then for any
] 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:
f ( x ) = 6 x 2 + 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 ,
] of linear way is equal to:
Thus, the work changes in the law: F ( x ) = 2 x 3 + 5 x .
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.