To set up the integrals necessary for calculating the mass, second moments, and radii of gyration of the given 2D object with density function $\rho(x, y) = x^2 y + y^3$, occupying the space inside the isosceles triangle connecting the points $(-3, 0)$, $(3, 0)$, and $(0, 9)$, we need to choose an appropriate coordinate system.

Justification for the Coordinate System

Given that the object occupies a triangular region, it is most convenient to use Cartesian coordinates. Cartesian coordinates are particularly suitable for problems involving triangles because these shapes are easily described by linear equations in the $xy$-plane.

Constraints in Cartesian Coordinates

The vertices of the triangle are $(-3, 0)$, $(3, 0)$, and $(0, 9)$. We need to find the equations of the lines forming the boundaries of the triangle.

1. Base of the triangle (horizontal line):

   • Equation: $y = 0$

2. Left side of the triangle:

   • Slope = $\frac{9 - 0}{0 - (-3)} = 3$
   • Equation: $y = 3(x + 3)$ or $y = 3x + 9$

3. Right side of the triangle:

   • Slope = $\frac{9 - 0}{0 - 3} = -3$
   • Equation: $y = -3(x - 3)$ or $y = -3x + 9$

The region of integration is bounded by:

• The line $y = 0$
• The line $y = 3x + 9$
• The line $y = -3x + 9$

Integral Setups

1. Mass $M$: $M = \iint\limits_D \rho(x, y) \, dA$ Here, $dA = dx \, dy$, and $\rho(x, y) = x^2 y + y^3$. The bounds of integration for $x$ are from $-3$ to $3$, and for $y$ are from $0$ to $-3x + 9$ (left side) or $3x + 9$ (right side).

   $M = \int_{-3}^{3} \int_{0}^{9 - 3|x|} (x^2 y + y^3) \, dy \, dx$

2. Second Moment about the $x$-axis $I_x$: $I_x = \iint\limits_D y^2 \rho(x, y) \, dA$

   $I_x = \int_{-3}^{3} \int_{0}^{9 - 3|x|} y^2 (x^2 y + y^3) \, dy \, dx$

3. Second Moment about the $y$-axis $I_y$: $I_y = \iint\limits_D x^2 \rho(x, y) \, dA$

   $I_y = \int_{-3}^{3} \int_{0}^{9 - 3|x|} x^2 (x^2 y + y^3) \, dy \, dx$

4. Product of Inertia $I_{xy}$: $I_{xy} = \iint\limits_D xy \rho(x, y) \, dA$

   $I_{xy} = \int_{-3}^{3} \int_{0}^{9 - 3|x|} xy (x^2 y + y^3) \, dy \, dx$

5. Radii of Gyration $(k_x, k_y)$: $k_x = \sqrt{\frac{I_x}{M}}, \quad k_y = \sqrt{\frac{I_y}{M}}$

Summary

The integrals set up for the mass, second moments, and radii of gyration in Cartesian coordinates provide a structured approach to solving the problem. Although we are not evaluating these integrals, the setup forms the foundation for further computation.

If you have any questions or need further details, please let me know.

Here are some questions you might want to explore next:

1. How do you derive the equations of lines forming the boundaries of a region?
2. What are the common applications of moments of inertia in engineering?
3. Can you provide an example of evaluating a double integral over a triangular region?
4. How does the density function $\rho(x, y)$ affect the moments of inertia?
5. What are the properties of the radii of gyration and their significance?

Tip: When setting up integrals for regions bounded by lines, it is helpful to sketch the region and clearly identify the equations of the boundaries to ensure correct limits of integration.