To find the sum of all integers between $\sqrt{27}$ and $\sqrt{99}$, we can first determine the integer values for both square roots:

$\sqrt{27} \approx 5.196 \quad \text{and} \quad \sqrt{99} \approx 9.95$

This means the integers between $\sqrt{27}$ and $\sqrt{99}$ are the integers from 6 to 9, inclusive.

The integers in this range are: 6, 7, 8, and 9.

Now, we calculate the sum:

$6 + 7 + 8 + 9 = 30$

So, the sum of all integers between $\sqrt{27}$ and $\sqrt{99}$ is 30.

Would you like further details or explanations? Here are some related questions to explore:

1. What is the sum of all integers between two other square roots, say $\sqrt{15}$ and $\sqrt{64}$?
2. How do square roots affect the range of integers considered?
3. Can you calculate the sum of integers in a similar range but including non-perfect square limits?
4. What is the formula to find the sum of an arithmetic series, like the one we just calculated?
5. How can you estimate the sum of integers in a given range without calculating exact values?

Tip: When calculating sums involving square roots, always approximate the square roots and identify the exact integer range before performing any calculations.