Let `f(x)=` maximum `{x^2, (1-x)^2, 2x(1 - x)}` where `x in [0, 1].` Determine the area of the region bounded by the curve `y=f(x)` and the lines `y = 0,x=0, x=1.`

Explanation

To solve the problem, we need to find the area bounded by the curve \( y = f(x) \) and the lines \( y = 0 \), \( x = 0 \), and \( x = 1 \), where \( f(x) = \max \{ x^2, (1-x)^2, 2x(1-x) \} \) for \( x \in [0, 1] \).

### Step-by-Step Solution:

1. **Identify the functions**:
We have three functions to consider:
- \( f_1(x) = x^2 \)
- \( f_2(x) = (1-x)^2 \)
...