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

Expert Answers

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You need to use absolute value definition such that:

`|x - 4| = x - 4 if x - 4 gt 0 =gt x gt 4`

`|x - 4| = 4 - x if x - 4 lt 0 =gt x lt 4`

Hence, you need to evaluate the definite integra over intervals [3,4] and [4,7] such that:

` int_3^7 (4 - |x - 4|)dx = int_3^4 (4 - 4 + x)dx + int_4^7(4 - x + 4)dx`  =>` int_3^7 (4 - |x - 4|)dx = int_3^4 xdx + int_4^7(8 - x)dx`

`int_3^7 (4 - |x - 4|)dx = x^2/2|_3^4 + 8x|_4^7 - x^2/2|_4^7`

`int_3^7 (4 - |x - 4|)dx = 16/2 - 9/2 + 56 - 32 - 49/2 + 16/2`

`int_3^7 (4 - |x - 4|)dx = 16 + 24 - 58/2`

`int_3^7 (4 - |x - 4|)dx = 40 - 29` `int_3^7 (4 - |x - 4|)dx = 11`

Hence, evaluating definite integral yields `int_3^7 (4 - |x - 4|)dx = 11.`

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