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