Help with this double integral

[Jackdeknipper]

Hi all,

Can one of you legends please check this exercise. I have seriously no idea what i am doing wrong. However the answer should be 8/9 according to the answers.
I know that i made a messy mistake by writing 8/9 instead of 8, however my final answer is in my opinion right.

Thanks a lot,

Cheers,

 

[pka]

Please try to post a readable image.

 

[Jackdeknipper]

Let's see of this works

 

[HallsofIvy]

If you make the substitution u= x+ y, v= x- y, then then dxdy= -(1/2)dudv, not dudv.

 

[Jackdeknipper]

If you make the substitution u= x+ y, v= x- y, then then dxdy= -(1/2)dudv, not dudv.

I used (1/2)dudv, because the determinant of the Jacobi Matrix is - 2. Therefore I use 1/abs(-2) as the correctionfactor.

Using - (1/2) would only lead to a negative number and of course integrals can not be negative.

 

[HallsofIvy]

Explain that more please. Why can't integrals be negative?

 

[Jackdeknipper]

Explain that more please. Why can't integrals be negative?

Sorry wrong interpretation. I meant because u and v are both non negative, the area under the curve has to be positive.

Futhermore I have learned that the correction factor is always the absolute value of the determinant of the Jacobi matrix. You are right that this determinant is - 2, however using the absolute value makes it 2. And this makes sense because u and v are non negative.