The area bounded by the curve `y={x}` with the x-axis from `x=pi` to `x=3.8` is `((pi)/(2)-a)(b-pi)` sq. units, then the value of `b-a` is equal to (where `{.}` denotes the fractional part function)

Answer

To solve the problem, we need to find the area bounded by the curve \( y = \{x\} \) (where \(\{x\}\) denotes the fractional part of \(x\)) and the x-axis from \( x = \pi \) to \( x = 3.8 \). We will then express this area in the form given in the question and find \( b - a \).

### Step 1: Understand the Function
The function \( y = \{x\} \) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means that the function gives the fractional part of \( x \). For example:
- For \( x = 3.2 \), \( \{3.2\} = 0.2 \)
- For \( x = 3.8 \), \( \{3.8\} = 0.8 \)

### Step 2: Determine the Area
We need to find the area under the curve from \( x = \pi \) to \( x = 3.8 \). The value of \( \pi \) is approximately \( 3.14 \), which means we are looking at the area from \( 3.14 \) to \( 3.8 \).

### Step 3: Identify the Interval
In the interval \( [\pi, 3.8] \):
- From \( x = \pi \) to \( x = 4 \), the function \( y = \{x\} \) will be a straight line starting from \( \{ \pi \} \) (which is \( \pi - 3 \)) and going up to \( 0 \) at \( x = 4 \).
- From \( x = 4 \) to \( x = 3.8 \), the function value will be \( 0.8 \).

### Step 4: Calculate the Area
The area can be calculated as the area of a trapezium formed by the points \( (\pi, \{\pi\}) \), \( (4, 0) \), and \( (3.8, 0.8) \).

1. **Base 1**: \( \{\pi\} = \pi - 3 \)
2. **Base 2**: \( 0.8 \)
3. **Height**: \( 3.8 - \pi \)

The area \( A \) of the trapezium is given by:
\[ A = \frac{1}{2} \times (\text{Base 1} + \text{Base 2}) \times \text{Height} \]
Substituting the values:
\[ A = \frac{1}{2} \times \left((\pi - 3) + 0.8\right) \times (3.8 - \pi) \]
\[ A = \frac{1}{2} \times \left(\pi - 2.2\right) \times (3.8 - \pi) \]

### Step 5: Compare with Given Area
According to the problem, the area is also given as:
\[ \left(\frac{\pi}{2} - a\right)(b - \pi) \]
We can equate the two expressions for area:
\[ \frac{1}{2} \times (\pi - 2.2) \times (3.8 - \pi) = \left(\frac{\pi}{2} - a\right)(b - \pi) \]

### Step 6: Identify Values of \( a \) and \( b \)
From the area calculation:
- We can simplify and find \( a \) and \( b \):
- \( a = 1.1 \)
- \( b = 3.8 \)

### Step 7: Calculate \( b - a \)
Now, we calculate \( b - a \):
\[ b - a = 3.8 - 1.1 = 2.7 \]

### Final Result
Thus, the value of \( b - a \) is:
\[ \boxed{2.7} \]