Ben is throwing darts at a circular target with diameter 10.  Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target.  Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target.  What is the expected number of points that Ben can earn from throwing a single dart?  (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)
1. **Define the problem and the scoring system:**
   - The target is a circle with a diameter of 10 units, hence a radius of 5 units.
   - Ben gets $\lceil 5 - x \rceil$ points for hitting a point $x$ units away from the center.
   - We need to calculate the expected number of points Ben can earn from a single dart throw.

2. **Divide the target into regions based on the scoring system:**
   - The scoring system is based on the distance from the center:
     - For $0 \leq x < 1$, Ben gets 5 points.
     - For $1 \leq x < 2$, Ben gets 4 points.
     - For $2 \leq x < 3$, Ben gets 3 points.
     - For $3 \leq x < 4$, Ben gets 2 points.
     - For $4 \leq x \leq 5$, Ben gets 1 point.

3. **Calculate the area of each region:**
   - The area of a circle with radius $r$ is $\pi r^2$.
   - The area of the region between two concentric circles with radii $r_1$ and $r_2$ is $\pi r_2^2 - \pi r_1^2$.

   Using this, we calculate the areas of the regions:
   - Region 1 ($0 \leq x < 1$): Area = $\pi (1^2 - 0^2) = \pi$
   - Region 2 ($1 \leq x < 2$): Area = $\pi (2^2 - 1^2) = 3\pi$
   - Region 3 ($2 \leq x < 3$): Area = $\pi (3^2 - 2^2) = 5\pi$
   - Region 4 ($3 \leq x < 4$): Area = $\pi (4^2 - 3^2) = 7\pi$
   - Region 5 ($4 \leq x \leq 5$): Area = $\pi (5^2 - 4^2) = 9\pi$

4. **Calculate the probability of hitting each region:**
   - The total area of the target is $\pi (5^2) = 25\pi$.
   - The probability of hitting each region is the area of the region divided by the total area.

   Thus, the probabilities are:
   - Region 1: $\frac{\pi}{25\pi} = \frac{1}{25}$
   - Region 2: $\frac{3\pi}{25\pi} = \frac{3}{25}$
   - Region 3: $\frac{5\pi}{25\pi} = \frac{5}{25}$
   - Region 4: $\frac{7\pi}{25\pi} = \frac{7}{25}$
   - Region 5: $\frac{9\pi}{25\pi} = \frac{9}{25}$

5. **Calculate the expected value:**
   - The expected value $E$ is given by the sum of the product of the points and the probabilities for each region.

   \[
   E = 5 \cdot \frac{1}{25} + 4 \cdot \frac{3}{25} + 3 \cdot \frac{5}{25} + 2 \cdot \frac{7}{25} + 1 \cdot \frac{9}{25}
   \]

   Simplifying this:

   \[
   E = \frac{5 \cdot 1 + 4 \cdot 3 + 3 \cdot 5 + 2 \cdot 7 + 1 \cdot 9}{25}
   \]

   \[
   E = \frac{5 + 12 + 15 + 14 + 9}{25}
   \]

   \[
   E = \frac{55}{25} = \frac{11}{5}
   \]

The final answer is $\boxed{\frac{11}{5}}$