Step 1: Idea:
Find the curved surface and the two ends separately, then add them.
Step 2: Curved surface:
Unroll the side into a rectangle of width $2\pi r$ and height $h$. Area $= 2 \times \dfrac{22}{7} \times 7 \times 11 = 44 \times 11 = 484$ cm$^2$.
Step 3: Two circular ends:
Each end has area $\pi r^2 = \dfrac{22}{7} \times 49 = 154$ cm$^2$. Two ends give $308$ cm$^2$.
Step 4: Add:
$484 + 308 = 792$ cm$^2$. This equals option 2. The value 748 would leave out part of the ends, and the larger options add too much.
Final Answer:
The total surface area is 792 cm$^2$.
\[ \boxed{792 \text{ cm}^2} \]