we consider the number of zeros of the integral $i(h) = oint_{gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. we prove that the number of zeros of $i(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

&nbsp;Let $ a_0&nbsp;(omega),&nbsp;a_1&nbsp;(omega),&nbsp;a_2&nbsp;(omega), dots,&nbsp;a_n&nbsp;(omega)$&nbsp;be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr,&nbsp;A)$. There are many known results for the expected number of real zeros of a polynomial&nbsp;$ a_0&nbsp;(omega) psi_0(x)+&nbsp;a_1&nbsp;(omega)psi_1 (x)+,&nbsp;a_2&nbsp;(omega)psi_2 (x)+...