The table shows the ages of some children in a playground. If the mean age is 8, calculate the value of x.

The mean ($\bar{A}$) is given by the sum of (age $\times$ frequency) divided by the sum of frequencies.

$$\bar{A} = \frac{\sum (A_i \times f_i)}{\sum f_i}$$

$$\sum (A_i \times f_i) = (6 \times 4) + (7 \times 10) + (8 \times 13) + (9 \times 8) + (10 \times x)$$ $$\sum (A_i \times f_i) = 24 + 70 + 104 + 72 + 10x$$ $$\sum (A_i \times f_i) = 270 + 10x$$

$$\sum f_i = 4 + 10 + 13 + 8 + x$$ $$\sum f_i = 35 + x$$

We are given that the mean age is 8.

$$8 = \frac{270 + 10x}{35 + x}$$

Multiply both sides by $(35 + x)$:

$$8(35 + x) = 270 + 10x$$ $$280 + 8x = 270 + 10x$$

Subtract $8x$ from both sides:

$$280 = 270 + 10x - 8x$$ $$280 = 270 + 2x$$

Subtract 270 from both sides:

$$280 - 270 = 2x$$ $$10 = 2x$$

Divide by 2:

$$x = \frac{10}{2}$$ $$x = 5$$

The value of $x$ is $\boxed{\text{5}}$.