absolute value integration problem

[lunarskull]

i am having a problem integrating this integral. the problem is:

the integral from 0 to 2 of "|(x^2)-x|

i know that the integral is (1/3)(x^3)-.5x^2 but i dunt no how to integrate an absolute value. i also no that u have to change the number in the integral sign. can someone start me on this problem?? plz and thank you

 

[Gene]

You have to split it into two integrals. If 0 < x < 1 then x²-x is negative so the absolute value is
x-x²
If 1 < x 2 then x²-x is positive and the absolute value is
x²-x so
do
integral from 0 to 1 of x-x² plus
integral from 1 to 2 of x²-x.
(When x=1 it is zero so that point doesn't contribute to the integral and using it twice doesn't hurt.)

 

[lunarskull]

thanks

o ok thanks. i understand now. just to be sure, is the answer 1?

 

[Gene]

Sure is.
Gene

 

[soroban]

Hello, lunarskull!

\(\displaystyle \L \int^{\;\;\;2}_0 |x^2\,-\,x|\,dx\)

This is the graph of \(\displaystyle y\:=\;x^2\,-\,x\)

    *   |               *
        |               :
     *  |              *:
        |               :
      * |             * :
        |               :
       *|            *  :
    ----*-----------*---+-
        | *       *
        |    * *

This is the graph of \(\displaystyle y\:=\:|x^2\,-\,x|\)

    *   |               *
        |               :
     *  |              *:
        |               :
      * |             * :
        |    * *        :
       *| *       *  *  :
    ----*-----------*---+-
        |