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The given integral is:

\[
\int_{-4}^{4} \int_{-\sqrt{16 - x^2}}^{\sqrt{16 - x^2}} \int_{-\sqrt{16 - x^2 - y^2}}^{\sqrt{16 - x^2 - y^2}} g(x, y, z) \, dz \, dy \, dx
\]

This represents a triple integral over a spherical region (the boundary limits suggest a circular cross-section in the \(xy\)-plane and spherical limits in the \(z\)-axis). You are asked to rewrite this integral using five different orders of integration.

Here are five possible orders of integration:

### 1. Order: \( dx \, dy \, dz \)
\[
\int_{-\sqrt{16 - y^2 - z^2}}^{\sqrt{16 - y^2 - z^2}} \int_{-\sqrt{16 - x^2}}^{\sqrt{16 - x^2}} \int_{-4}^{4} g(x, y, z) \, dx \, dy \, dz
\]

### 2. Order: \( dy \, dz \, dx \)
\[
\int_{-4}^{4} \int_{-\sqrt{16 - z^2}}^{\sqrt{16 - z^2}} \int_{-\sqrt{16 - x^2 - y^2}}^{\sqrt{16 - x^2 - y^2}} g(x, y, z) \, dy \, dz \, dx
\]

### 3. Order: \( dz \, dx \, dy \)
\[
\int_{-\sqrt{16 - y^2}}^{\sqrt{16 - y^2}} \int_{-\sqrt{16 - x^2 - z^2}}^{\sqrt{16 - x^2 - z^2}} \int_{-4}^{4} g(x, y, z) \, dz \, dx \, dy
\]

### 4. Order: \( dx \, dz \, dy \)
\[
\int_{-4}^{4} \int_{-\sqrt{16 - y^2}}^{\sqrt{16 - y^2}} \int_{-\sqrt{16 - x^2 - z^2}}^{\sqrt{16 - x^2 - z^2}} g(x, y, z) \, dx \, dz \, dy
\]

### 5. Order: \( dz \, dy \, dx \)
\[
\int_{-4}^{4} \int_{-\sqrt{16 - x^2}}^{\sqrt{16 - x^2}} \int_{-\sqrt{16 - x^2 - y^2}}^{\sqrt{16 - x^2 - y^2}} g(x, y, z) \, dz \, dy \, dx
\]

Would you like more details on the process of rewriting the integral in different orders?

Here are five related questions to deepen your understanding:
1. What is the geometric interpretation of the integration bounds in this spherical region?
2. How does the order of integration affect the ease of solving a triple integral?
3. Can the given integral be simplified using spherical coordinates?
4. What is the role of the function \( g(x, y, z) \) in this iterated integral?
5. How does one determine the appropriate limits when changing the order of integration?

**Tip**: Changing the order of integration can sometimes simplify the calculation, particularly when the bounds depend on one another.

Rewrite the following iterated integral using five different orders of integration: 
∫ from -4 to 4 ∫ from -√(16 − x²) to √(16 − x²) ∫ from -√(16 − x² − y²) to √(16 − x² − y²) g(x, y, z) dz dy dx.

Learn how to rewrite a triple integral using different orders of integration. This detailed guide covers the integral over a spherical region, including formulas, bounds, and theorems related to multivariable calculus. Ideal for students at the undergraduate level studying Calculus III or Multivariable Calculus.

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