Evaluating Triple Integral with Change of Variable: Help

[squenshl]

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

 

[tiny-tim]

Hi squenshl!

How do I evaluate the triple integral $$\int\int\int_G$$ x+y+z dV using a suitable change of variable where G is the region.

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

 

[tiny-tim]

x+y+z = (u+v+w)/2.

Woohoo! ​

 

[squenshl]

Cheers.