The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two
integers?
1. Let the two integers be \( x \) and \( y \). We are given two equations:
   \[
   x + y = 8
   \]
   \[
   x^2 + y^2 = 34
   \]

2. To find the product \( xy \), we start by squaring the first equation:
   \[
   (x + y)^2 = 8^2
   \]
   \[
   x^2 + 2xy + y^2 = 64
   \]

3. We already know from the second equation that:
   \[
   x^2 + y^2 = 34
   \]

4. Substitute \( x^2 + y^2 = 34 \) into the squared equation:
   \[
   34 + 2xy = 64
   \]

5. Solve for \( 2xy \):
   \[
   2xy = 64 - 34
   \]
   \[
   2xy = 30
   \]

6. Divide both sides by 2 to find \( xy \):
   \[
   xy = \frac{30}{2}
   \]
   \[
   xy = 15
   \]

The final answer is \( \boxed{15} \)