Evaluating Triple Integral with Change of Variable: Help

In summary, the conversation discusses evaluating the triple integral of x+y+z dV using a change of variables in a region G with given bounds. The suggested change of variables are u = x+y, v = y+z, and w = x+z. The conversation also mentions finding the determinant of the Jacobian and rewriting the integrand in terms of u, v, and w.
  • #1
squenshl
479
4
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.
 
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  • #2
Hi squenshl! :wink:
squenshl said:
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.

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 … ? :smile:
 
  • #3
Hint:

What does u+v+w equal, in terms of x+y+z?
 
  • #4
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.
 
Last edited:
  • #5
squenshl said:
x+y+z = (u+v+w)/2.

:biggrin: Woohoo! :biggrin:
 
  • #6
Cheers.
 

What is a triple integral?

A triple integral is an integral that involves three variables and is used to calculate the volume of a three-dimensional shape. It represents the sum of infinitesimal volumes over a three-dimensional region.

What is the change of variable method?

The change of variable method is a technique used to simplify the evaluation of integrals by substituting one variable for another. It is commonly used in triple integrals to transform the integrand into a simpler form.

What are the steps for evaluating a triple integral with change of variable?

The steps for evaluating a triple integral with change of variable are: 1) Choose appropriate variables for the transformation, 2) Determine the limits of integration in the new variables, 3) Calculate the new integrand using the Jacobian determinant, 4) Integrate the new function over the new limits of integration.

What is the Jacobian determinant?

The Jacobian determinant is a mathematical tool used in the change of variable method to calculate the transformation of variables. It is a matrix of partial derivatives that helps to convert integrals from one coordinate system to another.

What are some common applications of triple integrals with change of variable?

Triple integrals with change of variable have many applications in physics, engineering, and other scientific fields. They are used to calculate the mass, center of mass, moment of inertia, and other physical properties of three-dimensional objects. They are also used in probability and statistics to calculate the probability of events in three-dimensional space.

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