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.

Would you like a detailed solution for each integral, or further help with a specific part of this problem?

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.