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

1 Answer
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]."