The author considers an elliptic integral \(I_ \omega=\oint\omega\), where \(\omega\) is a real polynomial differential form of degree less than \(n\), and the integral is taken over a family of compact real ovals \(y^ 2+x^ 4-x^ 2-\lambda x=t\) on the plane. The question of the number of zeros of the integral as a function of the parameter \(t\) is resolved, depending on the degree of the integrand. The analogous question is considered for the integral \(I_ \omega\) taken over a family of compact real ovals \(y^ 2-x^ 4-x^ 2-\lambda x=t\) on the plane. Some results are also obtained for the maximal number of zeros of a hyperelliptic integral.