Question

Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

Hint:

Quick Tip: Break the integral at points where the absolute value expression changes its form, and handle each segment accordingly. This simplifies the computation.

Answer 1

To evaluate the integral, split the absolute value expressions based on the points where the expressions inside the absolute values change sign.
First, consider the piecewise forms of \( |x - 2| \) and \( |x - 4| \). For \( x \in [1, 5] \), the absolute values split as follows: \[ |x - 2| = \begin{cases} 2 - x, & \text{if } x < 2 \\ x - 2, & \text{if } x \geq 2 \end{cases} \] \[ |x - 4| = \begin{cases} 4 - x, & \text{if } x < 4 \\ x - 4, & \text{if } x \geq 4 \end{cases} \] Now, break the integral into intervals based on these points: \[ \int_1^5 \left( |x - 2| + |x - 4| \right) \, dx = \int_1^2 (2 - x + 4 - x) \, dx + \int_2^4 (x - 2 + 4 - x) \, dx + \int_4^5 (x - 2 + x - 4) \, dx \] Evaluate each integral: \[ \int_1^2 (6 - 2x) \, dx = [6x - x^2]_1^2 = (12 - 4) - (6 - 1) = 2 \] \[ \int_2^4 (2) \, dx = [2x]_2^4 = 8 - 4 = 4 \] \[ \int_4^5 (2x - 6) \, dx = [x^2 - 6x]_4^5 = (25 - 30) - (16 - 24) = -5 + 8 = 3 \] Thus, the total integral is: \[ 2 + 4 + 3 = 9 \] Hence, the value of the integral is: \[ \boxed{9} \]