Thread: double integration

integral integral |x-y|dydx where x is from 0 to 1 and y is from 0 to 1.
......
I don't know how to deal with this question because there is an absolute value.
I tried to split the range x to ( 0 to y and y to 1), but fail to get the correct answer. = =
Can anyone help me?
...
if you think my question typed is not clear, plz see the attached file.

Hey happymatthematics.

You need to split this into two integrals that are analytic.

Hint: When is x-y < 0 and when is x-y > 0? (This will help you find the limits to split the integral up)

this is my thought of next step...
I split 0<y<1 into 0<y<x and x<y<1,
but it doesn't work ;(
I got 2/3 but it is not the answer.
How to analyze?

I don't know the intuition to do this question.
What criteria should be fulfilled when I split the integral into two integrals?
Thank you!

Show us your working out in detail for the two integrals.

plz click the picture above to see my work.
Is this correct?
besides, this is my first time encountering this kind of questions,
I mean an absolute value inside an integral.
how to deal with a more complicated question?
can you give me a more complicated example with the intuition to solve this kind of problems?
sometimes I want to self-learn but cannot find similar questions in books.
I lack the intuitions to solve this kind of problems.
Thank you!

There is a on the third line which should be , but the line after that is ok.

You should look for problems where you can change the order of integration and do the integral both ways. So you should get the same result with y first as with x first. This problem is a special case where you had to divide the area into two parts.

- Hollywood