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To determine the average velocity of a fluid flow over a certain region, we need to consider the velocity profile and the spatial limits over which the flow occurs. In the context of an incompressible fluid flowing horizontally in the x-y plane, the average velocity, denoted as \( \overline{u} \), is found by integrating the velocity profile across the flow region and dividing this by the width of the region.
For our specific problem, this concept is represented by the formula: \[ \overline{u} = \frac{1}{h} \int_0^h u(y) \, dy \] Here, \( h \) is the upper bound of the region of interest, which is the maximum height in the y-direction of our flow region, and \( u(y) \) is the velocity profile that varies with \( y \).
The key steps in the average velocity calculation involve setting up the integral using the given velocity equation, calculating the integral, and then simplifying the result to find \( \overline{u} \). This approach ensures we accurately depict how the velocity changes across the fluid's flow path. Understanding this process is essential for various engineering applications, such as designing pipes or predicting fluid movement.

To find the average velocity of a fluid, integrating the velocity profile over the defined flow region is a pivotal step. This helps to understand how the fluid speed varies throughout that region, providing insight into the fluid's behavior and characteristics.
In our scenario, the velocity profile is provided as a function of y, \( u(y) = 30(y/h)^{1/2} \), where \( h \) is a constant. Setting up the integral involves placing this profile into the integral equation: \[ \overline{u} = \frac{1}{h} \int_0^h 30 \left( \frac{y}{h} \right)^{1/2} \, dy \]. Here, the function \( 30(y/h)^{1/2} \) needs to be integrated from \( 0 \) to \( h \). Remember, understanding this setup is crucial because the whole purpose is to assess the mean effect of the velocity across the span.
Notice how constants such as \( 30 \) and \( h^{-1/2} \) are factored out to simplify the integral. This simplification is integral (pun intended!) to solving the problem efficiently and getting a clear picture of the fluid's average velocity.

When you encounter expressions like \( y^{1/2} \) in integrals during fluid dynamics problems, the power rule for integration becomes valuable. The power rule states that \( \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \), useful for integrating polynomials and polynomial-like expressions.
Applying this rule to our expression, \( \int y^{1/2} \, dy \), transforms this into: \[ \left[ \frac{y^{3/2}}{3/2} \right]_0^h = \left[ \frac{2}{3} y^{3/2} \right]_0^h = \frac{2}{3} h^{3/2} \]. This result comes from substituting the upper and lower limits \( h \) and \( 0 \) into the integrated expression, resulting in \( \frac{2}{3} h^{3/2} \) because the zero term drops off.
The elegance of the power rule lies in its ability to handle fractional exponents easily and commonly appears in fluid dynamics scenarios. By mastering it, you're well-equipped to tackle complex velocity profiles and optimize fluid dynamic systems more effectively.