If the side of a water tank has width w(x*) at a depth of x* feet, set up an integral for the hydrostatic force on that side.  Assume the bottom of the tank is at a depth of 3 feet.

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.

If the side of a water tank has width w(x*) at a depth of x* feet, set up an integral for the hydrostatic force on that side. Assume the bottom of the tank is at a depth of 3 feet.

Discover how to set up an integral for calculating the hydrostatic force on the side of a water tank. This guide covers key concepts in calculus and hydrostatics, including pressure distribution, integration, and fluid physics, suitable for college-level students.

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