To solve the given integral over the region $E$, which is a box defined by $0 \leq x \leq 3$, $0 \leq y \leq 1$, and $0 \leq z \leq 1$, the integral can be expressed as:

$\iiint_E (x^2 + y^2 + z^2) \, dV = \int_0^3 \int_0^1 \int_0^1 (x^2 + y^2 + z^2) \, dz \, dy \, dx$

Step 1: Break the integral into components

The integrand can be split as: $\iiint_E (x^2 + y^2 + z^2) \, dV = \iiint_E x^2 \, dV + \iiint_E y^2 \, dV + \iiint_E z^2 \, dV$

Step 2: Evaluate each term

Term 1: $\iiint_E x^2 \, dV$

Since $x^2$ depends only on $x$: $\iiint_E x^2 \, dV = \int_0^3 x^2 \left( \int_0^1 \int_0^1 \, dz \, dy \right) dx$ Evaluate the inner integrals over $z$ and $y$: $\int_0^1 \int_0^1 \, dz \, dy = \int_0^1 [z]_0^1 \, dy = \int_0^1 1 \, dy = 1$ Thus: $\iiint_E x^2 \, dV = \int_0^3 x^2 \cdot 1 \, dx = \int_0^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^3 = \frac{27}{3} = 9$

Term 2: $\iiint_E y^2 \, dV$

Since $y^2$ depends only on $y$: $\iiint_E y^2 \, dV = \int_0^1 y^2 \left( \int_0^3 \int_0^1 \, dz \, dx \right) dy$ Evaluate the inner integrals over $z$ and $x$: $\int_0^3 \int_0^1 \, dz \, dx = \int_0^3 [z]_0^1 \, dx = \int_0^3 1 \, dx = 3$ Thus: $\iiint_E y^2 \, dV = \int_0^1 y^2 \cdot 3 \, dy = 3 \int_0^1 y^2 \, dy = 3 \left[ \frac{y^3}{3} \right]_0^1 = 3 \cdot \frac{1}{3} = 1$

Term 3: $\iiint_E z^2 \, dV$

Since $z^2$ depends only on $z$: $\iiint_E z^2 \, dV = \int_0^1 z^2 \left( \int_0^3 \int_0^1 \, dy \, dx \right) dz$ Evaluate the inner integrals over $y$ and $x$: $\int_0^3 \int_0^1 \, dy \, dx = \int_0^3 [y]_0^1 \, dx = \int_0^3 1 \, dx = 3$ Thus: $\iiint_E z^2 \, dV = \int_0^1 z^2 \cdot 3 \, dz = 3 \int_0^1 z^2 \, dz = 3 \left[ \frac{z^3}{3} \right]_0^1 = 3 \cdot \frac{1}{3} = 1$

Step 3: Combine the results

$\iiint_E (x^2 + y^2 + z^2) \, dV = \iiint_E x^2 \, dV + \iiint_E y^2 \, dV + \iiint_E z^2 \, dV = 9 + 1 + 1 = 11$

Final Answer:

$\iiint_E (x^2 + y^2 + z^2) \, dV = 11$

Let me know if you need any clarifications or further details! Here are some follow-up questions:

1. What if the boundaries of $x, y, z$ were different?
2. How would this change if the integrand was $x^2y^2z^2$?
3. Can this integral be solved using symmetry arguments?
4. How does the choice of integration order affect the computation?
5. How would this extend to higher-dimensional integrals?

Tip: When evaluating triple integrals, always check if the integrand can be split into independent functions of $x, y, z$. This simplifies the computation significantly!