How to solve that??#int_2^8|5-x|dx=?#

1 Answer
MattyMatty
May 7, 2018

#= 9#

Explanation:

#int_2^8 |5-x| dx = int_2^5 (5-x)dx + int_5^8 (x-5)dx#
#= [ 5x - x^2/2 + C1 ]_2^5 + [ x^2/2 - 5x + C2 ]_5^8#
#= 12.5 + C1 - 8 - C1 - 8 + C2 + 12.5 - C2#
#= 9#

#"In the first step we just apply the definition of |...| :"#

#|x| = {(-x, ", "x<=0),(x, ", "x >= 0):}#

#"So"#

#|5 - x| = { (x - 5, ", "5-x <= 0), (5 - x, ", "5-x >= 0):}#

#= {(x - 5, ", "x >= 5), (5 - x, ", "x <= 5):}#

#"So the limit case x=5 splits the integration interval up in two"#
#"parts : [2 ,5] and [5, 8]."#

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