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First, let us consider the pressure exerted by a fluid column. The pressure at any depth in a fluid is given by the equation: P = 蟻gh, where P is the pressure, 蟻 is the fluid density, g is the acceleration due to gravity, and h is the depth below the surface. As we can see, the pressure is directly proportional to the depth (h) and has no dependence on horizontal position.

If we were to integrate horizontally, we would irrespectively assume a constant depth at each point across the dam's face. This approach would not take into account the variations in pressure due to increasing depth as we go down the dam's face, making the integration inaccurate. On the other hand, if we integrate vertically, we can take into account the variations in pressure that arise due to the increasing depth of fluid along the dam's face. Since the pressure (and therefore the force) exerted on the dam's face is directly influenced by the fluid column's depth, integrating vertically allows us to capture the correct pressure distribution and therefore provides an accurate calculation of the force acting on the dam's face.

The acceleration due to gravity (g) acts vertically downward, which causes the pressure at a given depth to be greater than that at a higher point in the fluid column. This is the primary source of the pressure variation along the dam's face. Since the vertical direction is parallel to the gravitational force, integrating in the vertical direction correctly takes into account the influence of gravity on the force exerted on the dam's face.

In conclusion, to accurately calculate the force acting on a dam's face, it is crucial to integrate in the vertical direction. This is because the pressure exerted on the dam depends on the fluid column's depth, which increases as we move vertically along the dam's face. Integrating vertically allows us to account for this pressure variation, as well as the influence of gravity, leading to a correct calculation of the force on the dam's face. In contrast, integrating horizontally would not take into account the varying pressure along the dam's face and yield an inaccurate result.