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Pressure is a concept that helps us understand how force is applied over a specific area. In the context of fluids like water, pressure depends on how deep the fluid is. This is because the weight of the water above pushes downwards, increasing the amount of force at greater depths. The formula for hydrostatic pressure is given by:

• \( P = \rho g h \)

Here, \( P \) is the pressure, \( \rho \) is the density of the liquid (for water, it is 1000 kg/m³), \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the depth of the liquid.
Using this formula, one can calculate how much pressure is exerted at a particular depth within the aquarium by simply multiplying these values together.
Thus, as the depth of the water increases, so does the pressure. This increase in pressure leads directly to an increase in the force applied on the walls of the aquarium.

Integration is a powerful mathematical tool used to calculate total values such as force when they change with respect to another variable, like depth.
In our problem, we use integration to find the total force on the aquarium wall at different water depths. The force exerted by the water at any particular point depends on the pressure, which changes with depth.
To calculate the total force (\( F \)) experienced by a wall, we set up an integral of pressure over the height of the wall:

• \( F = \rho g \int_{0}^{h} x \cdot w \cdot dx \)

Here, \( x \) represents the depth variable, and \( w \) is the width of the wall.
For a change in the height of water from 2 m to 4 m, two separate integrals must be calculated. These integrals help determine the initial and new forces respectively.
This method shows how integration helps in calculating the total effect when there are varying amounts of pressure across a distance, like the height of an aquarium wall.

When solving physics problems, especially those involving hydrostatic force, following a structured approach helps to clearly work through the calculation stages.
Here are key steps used in the problem:

• Identify the core element of what needs to be found, such as the change in force when water depth changes in this case.
• Understanding concepts and formulas, like hydrostatic pressure \( P = \rho g h \) and how they apply to the scenario.
• Set up equations to represent the physical situation, such as integral equations for force.
• Perform integration calculations to solve the equations. This involves functional steps like setting limits for depth and integrating pressure over the established range.
• Finally, use basic arithmetic to find the increase in force using the results from your integrations.

By sequentially and peacefully working through these steps, complicated physics problems become manageable, and the principles become clearer.