  This paper develops a new integrated ball (pseudo)metric which provides an
intermediary between a chosen starting (pseudo)metric d and the L_p distance in
general function spaces. Selecting d as the Hausdorff or Fr\'echet distances,
we introduce integrated shape-sensitive versions of these supremum-based
metrics. The new metrics allow for finer analyses in functional settings, not
attainable applying the non-integrated versions directly. Moreover, convergent
discrete approximations make computations feasible in practice.
