Evaluate :`int_(0)^(8) |x-5|dx`

Answer

To evaluate the integral \( \int_{0}^{8} |x - 5| \, dx \), we will follow these steps:

### Step 1: Identify the points where the expression inside the absolute value changes sign.
The expression \( |x - 5| \) changes sign at \( x = 5 \). Therefore, we need to split the integral at this point.

### Step 2: Split the integral into two parts.
We can break the integral into two parts:
\[ \int_{0}^{8} |x - 5| \, dx = \int_{0}^{5} |x - 5| \, dx + \int_{5}^{8} |x - 5| \, dx \]

### Step 3: Determine the sign of \( x - 5 \) in each interval.
- For \( x \in [0, 5] \), \( x - 5 < 0 \) so \( |x - 5| = -(x - 5) = 5 - x \).
- For \( x \in [5, 8] \), \( x - 5 \geq 0 \) so \( |x - 5| = x - 5 \).

### Step 4: Rewrite the integral with the determined expressions.
Now we can rewrite the integral:
\[ \int_{0}^{8} |x - 5| \, dx = \int_{0}^{5} (5 - x) \, dx + \int_{5}^{8} (x - 5) \, dx \]

### Step 5: Evaluate the first integral \( \int_{0}^{5} (5 - x) \, dx \).
\[ \int_{0}^{5} (5 - x) \, dx = \left[ 5x - \frac{x^2}{2} \right]_{0}^{5} = \left( 5(5) - \frac{5^2}{2} \right) - \left( 5(0) - \frac{0^2}{2} \right) \]
Calculating this gives:
\[ = \left( 25 - \frac{25}{2} \right) - 0 = 25 - 12.5 = 12.5 \]

### Step 6: Evaluate the second integral \( \int_{5}^{8} (x - 5) \, dx \).
\[ \int_{5}^{8} (x - 5) \, dx = \left[ \frac{x^2}{2} - 5x \right]_{5}^{8} = \left( \frac{8^2}{2} - 5(8) \right) - \left( \frac{5^2}{2} - 5(5) \right) \]
Calculating this gives:
\[ = \left( \frac{64}{2} - 40 \right) - \left( \frac{25}{2} - 25 \right) = (32 - 40) - \left( 12.5 - 25 \right) \]
\[ = -8 - (-12.5) = -8 + 12.5 = 4.5 \]

### Step 7: Combine the results of both integrals.
Now we can combine the results:
\[ \int_{0}^{8} |x - 5| \, dx = 12.5 + 4.5 = 17 \]

### Final Answer:
Thus, the value of the integral is:
\[ \int_{0}^{8} |x - 5| \, dx = 17 \]