Let the three consecutive integers be represented as \( x \), \( x+1 \), and \( x+2 \).
The sum of these three integers can be expressed as:
\[ x + (x + 1) + (x + 2) = 228 \]
Simplifying the left side, we get:
\[ 3x + 3 = 228 \]
Now, we can subtract 3 from both sides:
\[ 3x = 228 - 3 \]
\[ 3x = 225 \]
Next, we divide both sides by 3 to solve for \( x \):
\[ x = \frac{225}{3} = 75 \]
Thus, the three consecutive integers are:
\[ 75, 76, \text{ and } 77 \]
The greatest of these integers is:
\[ \boxed{77} \]
To verify, we can check the sum of these integers:
\[ 75 + 76 + 77 = 228 \]
The sum is correct, confirming that the largest integer is indeed \( \boxed{77} \).