To set up an integral for the hydrostatic force on the side of the water tank, we'll follow these steps:

1. Understanding the Problem:

   • Let $x^*$ be the depth below the surface of the water (in feet).
   • The width of the tank at depth $x^*$ is given by $w(x^*)$.
   • The bottom of the tank is at a depth of 3 feet.
   • The hydrostatic pressure $P(x^*)$ at depth $x^*$ is given by $P(x^*) = \rho g x^*$, where:
     • $\rho$ is the density of water (approximately 62.4 lb/ft³ for fresh water).
     • $g$ is the acceleration due to gravity (approximately 32.2 ft/s²).

2. Differential Force:

   • Consider a thin horizontal strip at depth $x^*$ with thickness $\delta x^*$.
   • The area of this strip is $w(x^*) \delta x^*$.
   • The differential hydrostatic force $dF$ on this strip is: $dF = P(x^*) \times \text{Area of the strip} = \rho g x^* \times w(x^*) \delta x^*$ $dF = \rho g w(x^*) x^* \delta x^*$

3. Total Hydrostatic Force:

   • To find the total hydrostatic force $F$ on the side of the tank, integrate the differential force from the surface of the water (at $x^* = 0$) to the bottom of the tank (at $x^* = 3$ feet): $F = \int_{0}^{3} \rho g w(x^*) x^* \, dx^*$

4. Final Integral:

   • Substitute the values of $\rho$ and $g$ to get the integral for the hydrostatic force: $F = 62.4 \times 32.2 \int_{0}^{3} w(x^*) x^* \, dx^*$

Thus, the integral expression for the hydrostatic force on the side of the water tank is:

$F = 2008.48 \int_{0}^{3} w(x^*) x^* \, dx^*$

This integral will give you the total hydrostatic force on the side of the tank, assuming the width function $w(x^*)$ is known.

Would you like to go through the evaluation of this integral for a specific width function $w(x^*)$, or do you have any further questions?

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Related Questions:

1. How would the hydrostatic force change if the tank were deeper than 3 feet?
2. What if the width function $w(x^*)$ was constant across all depths?
3. How does the shape of the tank affect the width function $w(x^*)$?
4. How would the integral change if the tank contained a fluid other than water?
5. Can this integral be applied to a curved tank surface, and how would it change?

Tip: When dealing with hydrostatic forces, always consider the pressure distribution and how it varies with depth, as it will significantly impact the total force exerted on submerged surfaces.