Let's solve the expressions:
1. $$\frac{11}{14} \approx 0.7857$$
2. $$\sqrt{9} = 3$$
3. $$\sum_{i=2}^{6} i = 2 + 3 + 4 + 5 + 6 = 20$$
4. $$\log_4(14)$$. Since $$4^1 = 4$$ and $$4^2 = 16$$, $$\log_4(14)$$ is between 1 and 2. Let's estimate it to be approximately 1.9.
5. $$\int_{3}^{9} x dx = \frac{x^2}{2} \Big|_3^9 = \frac{9^2}{2} - \frac{3^2}{2} = \frac{81}{2} - \frac{9}{2} = \frac{72}{2} = 36$$
6. $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$
7. $$e^6 \approx 403.43$$
8. $$\frac{2\pi}{2} = \pi \approx 3.14$$
Ordering these from least to greatest:
$$\frac{11}{14} \approx 0.7857$$
$$\sqrt{9} = 3$$
$$\frac{2\pi}{2} = \pi \approx 3.14$$
$$\log_4(14) \approx 1.9$$
$$\sum_{i=2}^{6} i = 20$$
$$\int_{3}^{9} x dx = 36$$
$$5! = 120$$
$$e^6 \approx 403.43$$
Final order:
1. $$\frac{11}{14}$$ (0.7857)
2. $$\sqrt{9}$$ (3)
3. $$\frac{2\pi}{2}$$ (3.14)
4. $$\log_4(14)$$ (1.9)
5. $$\sum_{i=2}^{6} i$$ (20)
6. $$\int_{3}^{9} x dx$$ (36)
7. $$5!$$ (120)
8. $$e^6$$ (403.43)