1. **State the problem:** Calculate the sum of the integers from $i=5$ to $i=7$, i.e., compute $\sum_{i=5}^7 i$. 2. **Formula used:** The summation of consecutive integers from $a$ to $b$ can be calculated using the formula for the sum of an arithmetic series: $$\sum_{i=a}^b i = \frac{(b - a + 1)(a + b)}{2}$$ 3. **Apply the formula:** Here, $a=5$ and $b=7$. $$\sum_{i=5}^7 i = \frac{(7 - 5 + 1)(5 + 7)}{2} = \frac{3 \times 12}{2}$$ 4. **Simplify the expression:** $$\frac{3 \times 12}{2} = \frac{\cancel{3} \times 12}{\cancel{2}} = 3 \times 6 = 18$$ 5. **Interpretation:** The sum of the integers 5, 6, and 7 is 18. **Final answer:** $$\sum_{i=5}^7 i = 18$$