Why is There a - Sign in an Integral?

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The negative sign in front of the integral of (z2-z1)dA indicates that z1 is greater than z2, which is necessary for the volume element's height to remain positive. This suggests that z=0 is considered the fluid's surface, leading to z1 and z2 being negative values. The negative sign ensures that the force acts in the positive z direction. This interpretation aligns with the need for consistency in the mathematical representation of the physical scenario. Understanding this context clarifies the reasoning behind the negative sign in the integral.
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Homework Statement


why there is - sign in front of integral of (z2-z1)dA , is it a mistake ?

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In your picture, they define the height of the volume element (a positive number) as z1-z2. This means that z1 must be the larger number. I'm guessing that they must be thinking of z=0 as the surface of the fluid, making z1 and z2 negative numbers...so they have the negative sign out front to make the force act in the positive z direction. Just a guess anyway.
 
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