Change of variable

squenshl

How do I evaluate the triple integral [tex]\int\int\int_G[/tex] x+y+z dV using a suitable change of variable where G is the region
0 [tex]\leq[/tex] x+y [tex]\leq[/tex] 1, 2 [tex]\leq[/tex] y+z [tex]\leq[/tex] 3, 4 [tex]\leq[/tex] x+z [tex]\leq[/tex] 5.
I know to let u = x+y, v = y+z, w = x+z and I end up with the
det(jac) = |2| [tex]\Rightarrow[/tex] 1/det(jac) = |1/2|. But I'm stuck after that. Help.

tiny-tim

Hi squenshl!
Well, you've got the bounds, and you know how to rewrite the dV (from the Jacobian), so all you need is to rewrite x+y+z in terms of u v and w, which is … ?

arildno

Hint:

What does u+v+w equal, in terms of x+y+z?

squenshl

u+v+w = 2x+2y+2z = 2(x+y+z),
[tex]\Rightarrow[/tex] x+y+z = (u+v+w)/2.
Then just chuck that in. Is that right. Thanks.

tiny-tim

squenshl

Cheers.