https://file.aiquickdraw.com/mathbot/file/1729198619857ne2303u9.jpg

The given iterated integral in the image is:

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

This integral represents a triple integral with limits of integration in terms of \(x\), \(y\), and \(z\). The region of integration describes a portion of a cylinder (due to the bounds involving \(\sqrt{16 - x^2}\)).

To rewrite this integral in five different orders of integration, we must carefully change the bounds of integration while preserving the geometry of the region. Let’s explore the different possible orderings:

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

### 2. Order: \(dy \, dz \, dx\)
- \(z\) varies from \(x^2 + y^2\) to 16.
- \(y\) varies from \(-\sqrt{16 - x^2}\) to \(\sqrt{16 - x^2}\).
- For each fixed \(x\), integrate first with respect to \(y\) then with respect to \(z\).

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

### 3. Order: \(dz \, dx \, dy\)
- \(x\) varies from \(-\sqrt{16 - z}\) to \(\sqrt{16 - z}\), since \(z = 16 - x^2 - y^2\).
- \(z\) varies from \(x^2 + y^2\) to 16.

\[
\int_{-\sqrt{16}} \z_block_resize

Rewrite the following iterated integral using five different orders of integration: \[\int_{-4}^{4} \int_{-\sqrt{16 - x^2}}^{\sqrt{16 - x^2}} \int_{x^2 + y^2}^{16} g(x, y, z) dz dy dx \]

Explore how to rewrite the given iterated integral in five different orders of integration, using multivariable calculus concepts like triple integrals and cylindrical coordinates. Understand the application of Fubini's Theorem to change the order of integration while preserving the region of integration.

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