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The integral shown in the image is a triple integral of the function \( g(x, y, z) \), with the order of integration as \( dz \, dy \, dx \). The limits of integration for each variable are as follows:

- \( x \in [-4, 4] \)
- \( y \in \left[-\sqrt{16 - x^2}, \sqrt{16 - x^2}\right] \)
- \( z \in \left[x^2 + y^2, 16\right] \)

To rewrite this iterated integral in five different orders of integration, we need to consider all six possible orders for a triple integral:

1. **Original order**: \( dz \, dy \, dx \) (already provided).
2. **Alternative orders**: We will compute the remaining five orders: \( dz \, dx \, dy \), \( dy \, dz \, dx \), \( dy \, dx \, dz \), \( dx \, dz \, dy \), and \( dx \, dy \, dz \).

### Step-by-step transformations:

1. **\( dz \, dx \, dy \)**:
   - \( y \in [-4, 4] \)
   - \( x \in \left[-\sqrt{16 - y^2}, \sqrt{16 - y^2}\right] \)
   - \( z \in \left[x^2 + y^2, 16\right] \)

2. **\( dy \, dz \, dx \)**:
   - \( x \in [-4, 4] \)
   - \( z \in [x^2, 16] \)
   - \( y \in \left[-\sqrt{16 - x^2 - z}, \sqrt{16 - x^2 - z}\right] \)

3. **\( dy \, dx \, dz \)**:
   - \( z \in [0, 16] \)
   - \( x \in [-4, 4] \)
   - \( y \in \left[-\sqrt{16 - x^2}, \sqrt{16 - x^2}\right] \)

4. **\( dx \, dz \, dy \)**:
   - \( y \in [-4, 4] \)
   - \( z \in \left[y^2, 16\right] \)
   - \( x \in \left[-\sqrt{16 - y^2 - z}, \sqrt{16 - y^2 - z}\right] \)

5. **\( dx \, dy \, dz \)**:
   - \( z \in [0, 16] \)
   - \( y \in \left[-\sqrt{16 - z}, \sqrt{16 - z}\right] \)
   - \( x \in \left[-\sqrt{16 - y^2 - z}, \sqrt{16 - y^2 - z}\right] \)

These are the five alternative orders of integration for the triple integral.

Would you like further details on how each step was derived, or do you have any other questions?

Here are five follow-up questions to explore:

1. Can you explain the method for changing the order of integration in general terms?
2. How do the limits of integration change when switching the order of integration?
3. What is the significance of the bounds being dependent on other variables (e.g., \( \sqrt{16 - x^2} \))?
4. How does this problem relate to volumes under surfaces or regions in 3D space?
5. How would the integral be solved if a specific function \( g(x, y, z) \) were provided?

**Tip**: When changing the order of integration, visualize the region of integration in three dimensions to ensure the correct bounds for each variable.

Rewrite the following iterated integral using five different orders of integration.

Learn how to rewrite a given iterated integral in five different orders using Fubini's Theorem. Explore the bounds for x, y, and z and how they change when switching the order of integration. This problem is ideal for students studying multivariable calculus at the university level.

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