To find the center of mass for the region bounded by the ellipse equation:

$\frac{x^2}{4} + \frac{y^2}{9} = 1$

we can break it down step by step. The ellipse is centered at the origin $(0, 0)$, and the bounds are:

• $x$ ranges from $-2$ to $2$
• $y$ ranges from $-3$ to $3$.

Steps to Find the Center of Mass:

1. Equations for the Center of Mass (centroid): The center of mass in a region is given by the coordinates $(\bar{x}, \bar{y})$, where: $\bar{x} = \frac{1}{A} \int\int_R x \, dA$ $\bar{y} = \frac{1}{A} \int\int_R y \, dA$ where $A$ is the area of the region.

2. Area of the Ellipse: The area of an ellipse is given by: $A = \pi \cdot a \cdot b$ where $a$ and $b$ are the semi-major and semi-minor axes. For the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, $a = 2$ and $b = 3$, so: $A = \pi \cdot 2 \cdot 3 = 6\pi$

3. Symmetry Considerations: Since the ellipse is symmetric about both the $x$- and $y$-axes, the integrals involving $x$ and $y$ will both cancel out, meaning: $\bar{x} = 0, \quad \bar{y} = 0$ The center of mass of the region is at the origin $(0, 0)$.

Conclusion:

The center of mass for the region bounded by the ellipse is at the point: $(0, 0)$

Would you like further details on the integration process or have any questions?

Here are 5 related questions to expand on this:

1. How does symmetry affect the calculation of the center of mass?
2. What is the significance of the semi-major and semi-minor axes in an ellipse?
3. How would the center of mass change if the ellipse was not centered at the origin?
4. Can the center of mass be outside the region of a shape?
5. How is the area of an ellipse derived?

Tip: When dealing with symmetric regions, check for axis symmetry first—it can simplify your calculations significantly.