In this page we are going to integrate the differential expression of the hydrostatic pressure in order to calculate the pressure in a liquid at a given depth (*y* coordinate).

In the figure below a liquid contained in an open vessel is shown. Since the vessel is open to the atmosphere, the pressure p_{2} on its surface will be the atmospheric pressure:

The previously derived differential expression of pressure is given by:

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Integrating (assuming a constant density):

And finally, isolating p_{1} and substituting p_{2} for the **atmospheric pressure**:

The above expression explains why **the deeper you go under the sea, the greater the pressure of the water pushing down on you**. More precisely, for every 10 m you go down, the pressure increases by one atmosphere (1 atm approximately equals to 10 ^{5} Pa). To prove it, we just have to find the value of the term ρ g h, given that the density of seawater is approximately ρ = 1025 kg/m^{3}, g = 10 m/s^{2} and substituting h by 10 m:

From the expression above the so called ** Pascal’s law** (or ** Pascal’s principle**) can be stated:

*A pressure change at any point in a confined incompressible fluid is transmitted equally throughout the fluid.*

An interesting application of Pascal’s law is the **hydraulic press**. The hydraulic lift for automobiles shown schematically in the figure below is an example of a force multiplied by hydraulic press. It consists of two connected cylinders, one significantly bigger than the other, filled with an incompressible liquid:

Since an externally applied pressure is transmitted to all parts of the enclosed fluid, a small force applied on the left cylinder (represented in green), can be multiplied on the other side.

Mathematically:

Therefore, the larger the area, the larger the force on that side of the lift.

A **hydraulic brake** follows the same principle. Since the brake’s area is larger than that of the cylinder containing the fluid, a small force applied on the pedal is transformed into a larger one at the brake.