Question

Evaluate: \[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx. \]

Hint:

For absolute value functions, split the integral at points where the expression inside the modulus changes sign.

Answer 1

Step 1: Split the integral at critical points \( x = 1, 2, 3 \). \[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx = \int_1^2 (4 - x) \, dx + \int_2^3 x \, dx \] Step 2: Evaluate each integral separately. \[ \int_1^2 (4 - x) \, dx = \left[ 4x - \frac{x^2}{2} \right]_1^2 \] \[ = \left( 8 - 2 \right) - \left( 4 - \frac{1}{2} \right) = 6 - 3.5 = 2.5 \] \[ \int_2^3 x \, dx = \left[ \frac{x^2}{2} \right]_2^3 = \frac{9}{2} - \frac{4}{2} = \frac{5}{2} \] Step 3: Add the results. \[ \int_1^3 (|x - 1| + |x - 2| + |x - 3|) \, dx = 2.5 + 2.5 = 5 \] Final Answer: \[ \boxed{5} \]