Final answer:
The surface area of a closed cylinder is the sum of the curved surface area and the areas of the two circular ends. The formula to calculate this is 2πrh + 2πr², not just one of the components.
Step-by-step explanation:
To calculate the surface area of a closed cylinder, you need to consider both the curved surface area and the area of the two ends (circular tops). The curved surface area can be thought of as the area you would cover with paint if you rolled the cylinder out into a rectangle. Therefore, it is the perimeter of the base circle (which is 2πr) times the height (h), giving us the formula 2πrh for the curved surface area. Additionally, each of the ends of the cylinder is a circle with area πr2, and since there are two identical ends in a closed cylinder, the total area for both ends is 2πr2. To find the total surface area of the cylinder, you add these two areas together:
Surface area of a closed cylinder = Curved surface area + area of both ends
= 2πrh + 2πr2
Therefore, the correct option from the given choices is Option 4: 2πrh + 2πr2, which takes into account both the curved surface and the two circular ends of the cylinder.