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Let's go through each question one at a time.

### Question 1: Moment of Inertia about the x-axis
1. **Given Information**:
   - \( a = 3 \) in.
   - \( b = 2.5 \) in.
   - The curve is defined by \( y = k x^{1/2} \).

2. **Determine \( k \) in terms of \( a \) and \( b \)**:
   To find the value of \( k \), we use the given point \((a, b)\), which means \( y = b \) when \( x = a \):
   \[
   b = k \cdot a^{1/2} \Rightarrow k = \frac{b}{a^{1/2}}
   \]
   Substituting the given values,
   \[
   k = \frac{2.5}{\sqrt{3}}
   \]

3. **Calculate the moment of inertia about the x-axis (\( I_x \))**:
   The moment of inertia of an area with respect to the x-axis for this type of curve \( y = k x^{1/2} \) requires integrating the differential area \( y \, dx \) multiplied by the square of the distance from the x-axis. So we set up:
   \[
   I_x = \int_0^a y^2 \, dx = \int_0^a (k x^{1/2})^2 \, dx = \int_0^a k^2 x \, dx
   \]
   Substitute \( k = \frac{2.5}{\sqrt{3}} \) and solve the integral. 

### Question 2: Moment of Inertia about the y-axis
The process for the moment of inertia about the y-axis (\( I_y \)) involves integrating the square of the x-coordinates times the differential area. So:
\[
I_y = \int_0^a x^2 \, y \, dx = \int_0^a x^2 \cdot k x^{1/2} \, dx = \int_0^a k x^{5/2} \, dx
\]
Then substitute \( k = \frac{2.5}{\sqrt{3}} \) and evaluate the integral.

### Question 3: Moment of Inertia of an S-Shape Beam
For this question, you would typically look up the properties of the "S10 x 25.4" shape in a steel shapes table, which will directly give the moment of inertia about the X-X axis in inches to the fourth power (\( \text{in}^4 \)). Without access to this table here, I recommend checking a mechanical or structural engineering reference book or database for the exact value.

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### Related Questions for Further Practice:
1. How do you calculate the centroid of a region bounded by a curve?
2. What are the applications of moment of inertia in engineering?
3. How do you determine the area under a curve using integration?
4. What is the relationship between the moment of inertia and the bending stress in beams?
5. How does the shape of an area affect its moment of inertia?

### Tip:
Always ensure units are consistent when calculating moments of inertia—typically in inches or meters, depending on the problem context.

Find the moment of inertia of area of the shape below with respect to the x-axis and y-axis. a = 3 in, b = 2.5 in. y = k * x^(1/2). Also, determine the moment of inertia for an S-shape beam for designation S10 x 25.4 about the X-X axis.

Step-by-step guide to calculate the moment of inertia of a parabolic area with respect to the x-axis and y-axis, given a = 3 in and b = 2.5 in. Also covers determining the moment of inertia for an S10 x 25.4 standard steel beam about the X-X axis using standard tables.

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