Applications of Integration to Find Hydrostatic Pressure

Hydrostatic pressure arises from fluid weight at rest and increases proportionally with depth.

For a horizontal surface submerged at a fixed depth, the pressure is constant and equals the product of fluid density, acceleration due to gravity, and depth.

But for a vertical surface like a dam face, each point lies at a different depth, causing the pressure to increase with depth and creating a non-uniform force distribution.

To calculate the total hydrostatic force from this varying pressure, the submerged section is divided into thin horizontal strips. Each strip lies at a specific depth and has an area equal to its width times a small vertical thickness.

The pressure on each strip equals the fluid’s weight density times its depth below the surface, and the force on that strip equals pressure multiplied by area.

Summing these forces across all strips approximates the total force using a Riemann sum. As the strip thickness approaches zero, the sum becomes a definite integral.

This integral expresses the total hydrostatic force as pressure times the width function, integrated over the submerged depth.

This method is essential in civil engineering for designing efficient retaining structures.