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Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with depth, as the weight of the fluid above a certain point in the fluid causes a greater pressure at greater depth. Since the force exerted by the fluid must be perpendicular to the surface of the dam, hydrostatic pressure is responsible for the force acting on the dam's face.

To find the total force acting on the dam's face, we need to consider the varying hydrostatic pressure at different depths. Since the pressure changes vertically, due to the increase in depth (and thus increase in the gravitational force acting on the fluid), it becomes necessary to integrate the pressure with respect to depth to calculate the force acting on the dam's face. In other words, we need to add up the contributions of the pressure at each depth, which requires integration in the vertical direction.

Integrating in the horizontal direction would mean adding up the pressure contributions parallel to the surface of the dam. Since the hydrostatic pressure depends on the depth of the fluid, integrating in the horizontal direction would not yield the total force acting on the dam's face. There is no change in pressure in the horizontal direction, as the fluid is at rest, so integrating horizontally would not provide any useful information about the force on the dam. In conclusion, the force on the face of a dam is determined by integrating the hydrostatic pressure in the vertical direction because it is the gravity-induced pressure changes with depth that affect the force on the dam's surface. Integrating in the horizontal direction would not yield accurate results, as it ignores the varying contributions of pressure at different depths.