Abstract
A kind of fuzzy distance gives a new definition of the definite integral of fuzzy valued function linearly generated by structural element. Then the new definition is used to study the basic properties of the definite integral of fuzzy valued function linearly generated by structural elements defined on the interval [a, b]. They are Newton Leibniz formula, addition together with multiplication, interval additivity, boundedness, local protection and the first mean value theorem for integrals. After that, By studing some theorems within Darboux sum definition of fuzzy valued function linearly generated by structural elements, The first integrable condition and the second integrable condition of fuzzy valued functions for linear generation of structural elements on [a, b] was given. Meanwhile, investigate the integrability of fuzzy valued function linearly generated by structural elements which is continuous or bounded on [a, b].
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