The problem in the image asks you to evaluate the following integral:

$\iiint_E (xy + z^2) \, dV$ where $E = \{ (x, y, z) \mid 0 \leq x \leq 3, 0 \leq y \leq 2, 0 \leq z \leq 5 \}$.

To solve this, we need to express the integral in terms of limits for the variables $x$, $y$, and $z$ based on the given region $E$. The bounds for $x$, $y$, and $z$ are all provided, which suggests using triple integration. The order of integration can be done in three possible ways.

Step-by-step approach:

1. For the first order of integration (dx, dy, dz): $\int_{z=0}^5 \int_{y=0}^2 \int_{x=0}^3 (xy + z^2) \, dx \, dy \, dz$

2. For the second order of integration (dy, dz, dx): $\int_{x=0}^3 \int_{z=0}^5 \int_{y=0}^2 (xy + z^2) \, dy \, dz \, dx$

3. For the third order of integration (dz, dx, dy): $\int_{y=0}^2 \int_{x=0}^3 \int_{z=0}^5 (xy + z^2) \, dz \, dx \, dy$

Would you like me to help you solve one of these integrals in detail, or would you prefer more guidance on how to approach the calculation?