Solving Integral of x|x| from -1 to 1

[olds442]

Homework Statement

integral of x|x|dx.. [-1,1]

The Attempt at a Solution

when graphing it, it is even (looks like the cubic function) and would be 0... but I am having problems convincing myself of this. i checked it out using my calculator, and it just gives back x|x|.. i know if you do the integral of abs x you would split it up into positive and negative parts and all that.. but this is obviously different since it goes below y=0...

 

[ptr]

You would still split it up and add the integral for x>0 up with the one for x<0.

 

[dynamicsolo]

The Attempt at a Solution

when graphing it, it is even (looks like the cubic function) and would be 0...

I think you mean it's an odd function. And the definite integral from -a to a should give you zero.

but I am having problems convincing myself of this. i checked it out using my calculator, and it just gives back x|x|.. i know if you do the integral of abs x you would split it up into positive and negative parts and all that.. but this is obviously different since it goes below y=0...

If you use the definition for absolute value,

|x| = x for x => 0 , -x for x < 0 ,

then your integrand is

x·|x| = x^2 for x => 0 , -(x^2) for x < 0 .

That will explain the graph and the result for your definite integral.