Finding Volume triple integral by change of variables

In summary, the formula for finding volume using triple integrals by change of variables is ∭∭∭ f(x,y,z) dV = ∭∭∭ f(u,v,w) |J(u,v,w)| dudvdw, where f(x,y,z) represents the function, J(u,v,w) is the Jacobian determinant of the transformation, and dudvdw are the differentials of the new variables u,v,w. The purpose of using a change of variables is to simplify the integrals and make them easier to solve, and the steps involved include identifying the region, choosing a suitable transformation, finding the Jacobian determinant, rewriting the integral, and evaluating it. Common transformations used include polar,
  • #1
nautolian
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Homework Statement



∫∫∫V 9z2 dxdydz, where V is the solid defined by:

-1≤x+y+3z≤1, 1≤2y-z≤7, -1≤x+y≤1

Homework Equations


The Attempt at a Solution


I did this using the bounds, 1/3(-x-y-1)<=z<=(1/3)(-x-y+1), -x-1<=y<=1-x, -3/2<=x<=1/2 but I think the answer is wrong is there a better way to do this with change of variables? Thanks, any help would be greatly appreciated.
 
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  • #2
Well, u = x+y+3z, v = 2y-z, w=x+y would surely give constant limits.
 

FAQ: Finding Volume triple integral by change of variables

What is the formula for finding volume using triple integrals by change of variables?

The formula for finding volume using triple integrals by change of variables is:

∭∭∭ f(x,y,z) dV = ∭∭∭ f(u,v,w) |J(u,v,w)| dudvdw

Where f(x,y,z) represents the function, J(u,v,w) is the Jacobian determinant of the transformation, and dudvdw are the differentials of the new variables u,v,w.

What is the purpose of using a change of variables in finding volume with triple integrals?

The purpose of using a change of variables in finding volume with triple integrals is to simplify the integrals and make them easier to solve. By replacing the original variables with new ones, the limits of integration can also be changed to better fit the shape of the region being integrated over, resulting in a simpler and more efficient calculation of volume.

What are the steps involved in using change of variables to find volume with triple integrals?

The steps involved in using change of variables to find volume with triple integrals are:

1. Identify the region to be integrated over and determine the limits of integration for each variable.

2. Choose a suitable transformation for the variables that will simplify the integrals and make the limits of integration more manageable.

3. Find the Jacobian determinant of the transformation.

4. Rewrite the original integral in terms of the new variables and the Jacobian determinant.

5. Evaluate the integral to find the volume.

What are some common transformations used in finding volume with triple integrals by change of variables?

Some common transformations used in finding volume with triple integrals by change of variables are polar, cylindrical, and spherical coordinates. These transformations are especially useful for integrating over regions with circular or spherical symmetry.

Other common transformations include transformations to Cartesian coordinates, which can simplify integrals over rectangular or parallelepiped regions.

What are some tips for choosing an appropriate transformation in finding volume with triple integrals by change of variables?

When choosing a transformation for finding volume with triple integrals by change of variables, it is important to consider the shape and symmetry of the region being integrated over. The transformation should also be chosen in such a way that the limits of integration become easier to work with. Additionally, it is important to choose a transformation that preserves the orientation of the region, as this will ensure that the resulting volume is accurate.

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