The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membership value of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is present has been well established. Most of the integral inequalities studied in the fuzzy integration context normally consider conditions such as monotonicity or comonotonicity. In this paper, we are trying to extend the fuzzy integrals to the concept of concavity. It is shown that the Hermite-Hadamard integral inequality for concave functions is not satisfied in the case of fuzzy integrals. We propose upper and lower bounds on the fuzzy integral of concave functions. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.