Step 1: Reframing using distance-from-point intuition:
Each \(|x-k|\) is the distance from \(x\) to \(k\); summing three such distances and integrating is equivalent to computing three separate simple absolute-value integrals and adding them.
Step 2: Splitting into three separate standard integrals:
\(\displaystyle\int_1^4|x-1|dx+\int_1^4|x-2|dx+\int_1^4|x-3|dx\), each evaluated independently by splitting at its own single breakpoint.
Step 3: Evaluating each one:
\(\int_1^4|x-1|dx=\int_1^4(x-1)dx=\tfrac{9}{2}\) (no sign change, \(x\ge1\) throughout). \(\int_1^4|x-2|dx=\int_1^2(2-x)dx+\int_2^4(x-2)dx=\tfrac12+2=\tfrac52\). \(\int_1^4|x-3|dx=\int_1^3(3-x)dx+\int_3^4(x-3)dx=2+\tfrac12=\tfrac52\).
Step 4: Summing the three results:
Total \(=\tfrac92+\tfrac52+\tfrac52=\tfrac{9+5+5}{2}=\tfrac{19}{2}\), matching the piecewise method.
Final Answer:
\[ \boxed{\dfrac{19}{2}} \]