# Evaluate the definite integral of the algebraic function. 7 (4 − |x − 4|) dx 3

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`int^(7)_3 (4 - |x - 4|) dx`

The definition of absolute value is: `|a|={(a if agt=0),(-a if alt0):}`

so |x-4| = x-4 if x>=4, and |x-4|=-(x-4) if x<4 so we need to split our integral into two pieces.

`int^(7)_3 (4-|x-4|) dx = int^(7)_4 (4 - (x - 4)) dx + int^(4)_3 (4 - (-(x - 4))) dx`

`= int^7_4 (8-x) dx + int^4_3 (x) dx = [8x - x^2/2]^7_4 + [x^2/2]^4_3`

`= 56 - 49/2 - (32 - 16/2) + 16/2 - 9/2 = 56 - 49/2 - 24 + 8 - 9/2 = 11`

So

`int^7_3 (4-|x-4|)dx = 11`