In this work, some new kinds of nonlinear power integral inequalities in two-dimensional cases are introduced. With advanced analytical mathematical methods, many estimations are obtained in order to generalize to higher order, well known results in the literature for $p=1$ and $p\in (0,1]$. As an example, we apply these results to determine the boundedness of solutions of nonlinear bidimensional Volterra type integral equations for $p>1$.