Final answer:
To express the hydrostatic force against one side of the plate as an integral, use the concept of pressure and integrate it over the surface area of the plate. The hydrostatic force is given by F = pghA, where p is the density of the fluid, g is the acceleration due to gravity, h is the height of the fluid column, and A is the area of the plate in contact with the fluid.
Step-by-step explanation:
To express the hydrostatic force against one side of the plate as an integral, we can use the concept of pressure and integrate it over the surface area of the plate.
The hydrostatic force is given by the equation F = pghA, where p is the density of the fluid, g is the acceleration due to gravity, h is the height of the fluid column, and A is the area of the plate in contact with the fluid. We can express this as an integral by integrating the pressure over the area of the plate.
Let's assume the plate has dimensions L and W, and is located at a depth d in the fluid. The pressure at a depth y in the fluid is given by p = p0 + pgy, where p0 is the pressure at the surface of the fluid.
Substituting this into the equation for the hydrostatic force, we get: F = ∫[p0 + pgy] dA = ∫[p0 + pgy] dxdy
Where the integral is taken over the area of the plate.