Evaluate : `int_(4)^(5)|x-5|dx`

Answer

To evaluate the integral \(\int_{4}^{5} |x - 5| \, dx\), we will follow these steps:

### Step 1: Analyze the absolute value function
The expression \(|x - 5|\) can be simplified based on the value of \(x\):
- For \(x < 5\), \(|x - 5| = 5 - x\).
- For \(x = 5\), \(|x - 5| = 0\).

Since our limits of integration are from 4 to 5, we will use \(|x - 5| = 5 - x\) for the entire interval \([4, 5)\).

### Step 2: Set up the integral
We can rewrite the integral using the simplified expression:
\[ \int_{4}^{5} |x - 5| \, dx = \int_{4}^{5} (5 - x) \, dx \]

### Step 3: Integrate the function
Now we will integrate \(5 - x\):
\[ \int (5 - x) \, dx = 5x - \frac{x^2}{2} + C \]

### Step 4: Evaluate the definite integral
Now we will evaluate this integral from 4 to 5:
\[ \left[ 5x - \frac{x^2}{2} \right]_{4}^{5} \]

Calculating at the upper limit \(x = 5\):
\[ 5(5) - \frac{(5)^2}{2} = 25 - \frac{25}{2} = 25 - 12.5 = 12.5 \]

Calculating at the lower limit \(x = 4\):
\[ 5(4) - \frac{(4)^2}{2} = 20 - \frac{16}{2} = 20 - 8 = 12 \]

### Step 5: Subtract the results
Now we subtract the result at the lower limit from the result at the upper limit:
\[ 12.5 - 12 = 0.5 \]

Thus, the value of the integral is:
\[ \int_{4}^{5} |x - 5| \, dx = 0.5 \]

### Final Answer
\[ \int_{4}^{5} |x - 5| \, dx = \frac{1}{2} \]

---