If f ( x )=| x -1|+| x -2|+| x -3|, x [1,4] , then ₁^4 f ( x ) dx =

Step-by-step solution

aligned & ₁^4 f(x) d x= ₁^4[|x-1|+|x-2|+|x-3| d x & = ₁^2(x-1)+(2-x)+(3-x) d x+ ₂^3[(x-1)+(x-2)+(3-x)] d x & + ₃^4[(x-1)+(x-2)+(x-3)] d x & = ₁^2(4-x) d x+ ₂^3 x d x+ ₃^4(3 x-6) d x aligned aligned & = [4 x- x^2 2 ]₁^2+ [ x^2 2 ]₂^3+ [ 3 x^2 2 -6 x ]₃^4 & = [(8-2)- (4- 1 2 ) ]+ [ ( 9 2 -2 ) ]+ [(24-24)- ( 27 2 -18 ) ] & = (2+ 1 2 )+ ( 5 2 )+ ( 9 2 )= 19 2 aligned