1. Given data

• Room dimensions: $4\,\text{m} \times 4\,\text{m} \times 3\,\text{m}$
• Volume: $V = 4 \times 4 \times 3 = 48\,\text{m}^3$
• Pressure: $P = 1\,\text{atm} \approx 1.013 \times 10^5\,\text{Pa}$
• Air is treated as a diatomic ideal gas.

2. Internal energy formula for diatomic gas

For an ideal gas, $U = \frac{f}{2}nRT$ where $f$ is the degrees of freedom.

For a diatomic gas at ordinary temperature, $f=5$. Hence, $U = \frac{5}{2}nRT$

Using the ideal gas law, $nRT = PV$ so, $U = \frac{5}{2}PV$

3. Substitute values

First calculate $PV$: $PV = (1.013 \times 10^5)(48)$ $PV = 48.624 \times 10^5$ $PV = 4.8624 \times 10^6\,\text{J}$

Now, $U = \frac{5}{2} \times 4.8624 \times 10^6$ $U = 2.5 \times 4.8624 \times 10^6$ $U = 12.156 \times 10^6\,\text{J}$

4. Required integer

The question asks for $\underline{\hspace{2cm}} \times 10^6\,\text{J}$ So the coefficient is approximately $12.156 \approx 12$

5. Final answer

$U \approx 12 \times 10^6\,\text{J}$ So the required integer is: $\boxed{12}$