To evaluate the definite integral \( \int_{3}^{5} |x-4| \, dx \), we need to handle the absolute value function carefully by understanding where the expression inside the absolute value changes sign.
The expression inside the absolute value is \( x - 4 \). This expression changes sign at \( x = 4 \). Therefore, we break the integral at this point:
We split the integral \(\int_{3}^{5} |x-4| \, dx\) into two parts:
\(\int_{3}^{5} |x-4| \, dx = \int_{3}^{4} (4 - x) \, dx + \int_{4}^{5} (x - 4) \, dx\)
Let's evaluate these two integrals separately.
First Integral: \(\int_{3}^{4} (4 - x) \, dx\)
Compute the antiderivative: The integral of \(4-x\) is \((4x - \frac{x^2}{2})\).
Evaluate from 3 to 4:
\[ \left[4x - \frac{x^2}{2}\right]_{3}^{4} = \left(4(4) - \frac{4^2}{2}\right) - \left(4(3) - \frac{3^2}{2}\right) \] \[ = \left(16 - 8\right) - \left(12 - 4.5\right) \] \[ = 8 - 7.5 = 0.5 \]
Second Integral: \(\int_{4}^{5} (x - 4) \, dx\)
Compute the antiderivative: The integral of \(x-4\) is \(\left(\frac{x^2}{2} - 4x\right)\).
Evaluate from 4 to 5:
\[ \left[\frac{x^2}{2} - 4x\right]_{4}^{5} = \left(\frac{5^2}{2} - 4(5)\right) - \left(\frac{4^2}{2} - 4(4)\right) \] \[ = \left(\frac{25}{2} - 20\right) - \left(8 - 16\right) \] \[ = \left(12.5 - 20\right) - \left(-8\right) \] \[ = -7.5 + 8 = 0.5 \]
Add the Results of Both Integrals:
\(0.5 + 0.5 = 1\)
Thus, the value of the definite integral \( \int_{3}^{5} |x-4| \, dx \) is 1, which seems to suggest a discrepancy since the expected answer is provided as 2. Upon rechecking, it confirms that the calculations imply a conceptual miscalculation in step understanding which showed nominal result pre-discussed. Hence, this computed answer is accurate as per logical structure.