We are asked to find the center of mass of a thin plate bounded by the x-axis and the curve $y = \frac{2}{x}$, for $1 \leq x \leq 4$, with a density function given as $\delta(x) = x^2$.

To calculate the center of mass $(\bar{x}, \bar{y})$, we will use the following formulas:

1. Total Mass (M): $M = \int_{1}^{4} \delta(x) y(x) \, dx = \int_{1}^{4} x^2 \cdot \frac{2}{x} \, dx = \int_{1}^{4} 2x \, dx$

2. $\bar{x}$ (center of mass in the x-direction): $\bar{x} = \frac{1}{M} \int_{1}^{4} x \delta(x) y(x) \, dx = \frac{1}{M} \int_{1}^{4} x \cdot x^2 \cdot \frac{2}{x} \, dx = \frac{1}{M} \int_{1}^{4} 2x^2 \, dx$

3. $\bar{y}$ (center of mass in the y-direction): $\bar{y} = \frac{1}{2M} \int_{1}^{4} \delta(x) [y(x)]^2 \, dx = \frac{1}{2M} \int_{1}^{4} x^2 \cdot \left(\frac{2}{x}\right)^2 \, dx = \frac{1}{2M} \int_{1}^{4} \frac{4}{x^2} x^2 \, dx$

Let me now compute the total mass $M$ and the coordinates $\bar{x}$ and $\bar{y}$.The center of mass of the thin plate is located at:

• Total mass $M = 15$
• $\bar{x} = \frac{14}{5} = 2.8$
• $\bar{y} = \frac{2}{5} = 0.4$

Would you like further details or clarification on any of these steps?

Here are five related questions for further exploration:

1. How does the center of mass change if the density function is uniform?
2. What happens to the center of mass if the curve is shifted or modified?
3. Can you calculate the moment of inertia of the plate around the x-axis?
4. How would the center of mass change for a different region or limits of integration?
5. How do the formulas for the center of mass change in three dimensions?

Tip: When calculating the center of mass for non-uniform densities, always check how the density function varies with respect to the coordinates for accuracy.