Hydrostatic Pressure

Calculus 2 · Axiom Academy

Pressure grows with depth, so the force on a submerged wall is an integral — a sum of thin horizontal strips, each pushed harder than the one above it. Inside a fluid at rest, the pressure at depth h is the weight of the water column standing above you, spread over each unit of area. It rises linearly with depth: Pressure is proportional to depth h Force per unit area, in pascals (Pa = N/m²). Mass per unit volume. Fresh water: kg/m³. Gravitational acceleration, m/s². Distance straight down below the surface, in meters. On a small flat-on-the-bottom patch at a single depth, force is easy: . But a dam stands vertically . Its top sits just below the surface where pressure is tiny; its base is deep where pressure is large. A single pressure can't describe the whole face. A strip at depth h with thickness dh on a wall of width w has area , so the force it feels is . The arrows fan out longer as the strips go deeper — that growing push is exactly what we have to sum. 3. Add the Strips: the Integral Summing every strip's force from the surface (h=0) down to the base (h=H) turns the sum into an integral. Because , g , and w are constants, only h is left under the sign: Pressure on a strip at depth h : The triangular pressure profile is the whole story: pressure runs from 0 at the top to at the base, so the total force is the area of that triangle times the width — which is precisely .

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