To rewrite the given triple integral in the order $dx \, dz \, dy$, we need to adjust the limits of integration accordingly. The original integral is:

$\iiint_D g(x, y, z) \, dz \, dy \, dx$

with the bounds:

$\int_{-3}^{3} \int_{-\sqrt{9 - x^2}}^{\sqrt{9 - x^2}} \int_{x^2 + y^2}^{9} g(x, y, z) \, dz \, dy \, dx$

Step 1: Understanding the Region

1. $x$ Limits: $x$ ranges from $-3$ to $3$.
2. $y$ Limits: For a fixed $x$, $y$ ranges from $-\sqrt{9 - x^2}$ to $\sqrt{9 - x^2}$. This describes a circular disk in the $xy$-plane with radius 3.
3. $z$ Limits: $z$ ranges from $x^2 + y^2$ (a paraboloid) to 9 (a plane).

Step 2: Rewriting the Integral

Now, we switch the order to $dx \, dz \, dy$. We need to identify how $x$ varies with respect to $z$ and $y$:

1. $y$ Limits: $y$ still ranges from $-3$ to $3$.
2. $z$ Limits: For a fixed $y$, $z$ ranges from $y^2$ (when $x = 0$) up to 9.
3. $x$ Limits: For fixed $y$ and $z$, $x$ ranges between $-\sqrt{z - y^2}$ and $\sqrt{z - y^2}$. This comes from solving $x^2 + y^2 = z$ for $x$.

Step 3: Writing the Integral

The integral becomes:

$\int_{-3}^{3} \int_{y^2}^{9} \int_{-\sqrt{z - y^2}}^{\sqrt{z - y^2}} g(x, y, z) \, dx \, dz \, dy$

This is the equivalent form of the integral in the order $dx \, dz \, dy$.

Would you like to go over any details or ask further questions?

1. How do you determine the limits when changing the order of integration?
2. Can the limits change if the region is described differently, such as in polar coordinates?
3. What is the role of symmetry in simplifying triple integrals?
4. How can one visualize the regions bounded by different limits in 3D?
5. What happens if the function $g(x, y, z)$ is specifically given as a simple polynomial?

Tip: When changing the order of integration, sketching the region helps in identifying the new limits correctly.