Change of Variables in Triple Integrals


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Change of Variables in Triple Integrals

As with double integrals, triple integrals can often be easier to evaluate by making the change of variables. This allows to simplify the region of integration or the integrand.

Let a triple integral be given in the Cartesian coordinates \(x, y, z\) in the region \(U:\)

\[\iiint\limits_U {f\left( {x,y,z} \right)dxdydz} .\]

We need to calculate this integral in the new coordinates \(u, v, w.\) The relationship between the old and new coordinates is given by

\[x = \varphi \left( {u,v,w} \right),\;\; y = \psi \left( {u,v,w} \right),\;\; z = \chi \left( {u,v,w} \right).\]

It is supposed here that the following conditions are satisfied:

  1. The functions \(\varphi, \psi, \chi\) are continuous together with their partial derivatives;
  2. There's a single valued relation between points of the region of integration \(U\) in the \(xyz\)-space and points of the region \(U'\) in the \(uvw\)-space;
  3. The Jacobian of transformation \(I\left( {u,v,w} \right)\) equal to
    \[ I\left( {u,v,w} \right) = \frac{{\partial \left( {x,y,z} \right)}}{{\partial \left( {u,v,w} \right)}} = \left| {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial u}}}&{\frac{{\partial x}}{{\partial v}}}&{\frac{{\partial x}}{{\partial w}}}\\ {\frac{{\partial y}}{{\partial u}}}&{\frac{{\partial y}}{{\partial v}}}&{\frac{{\partial y}}{{\partial w}}}\\ {\frac{{\partial z}}{{\partial u}}}&{\frac{{\partial z}}{{\partial v}}}&{\frac{{\partial z}}{{\partial w}}} \end{array}} \right|,\]
    is non-zero and keeps a constant sign everywhere in the region of integration \(U.\)

Then the formula for change of variables in triple integrals is written as

\[\iiint\limits_U {f\left( {x,y,z} \right)dxdydz} = \iiint\limits_{U'} {f\left( {\varphi,\psi,\chi} \right) \left|{I\left( {u,v,w} \right)} \right|dudvdw}.\]

Here \(\varphi,\psi,\chi\) are functions of the variables \({u,v,w}\) and \(\left| {I\left( {u,v,w} \right)} \right|\) means the absolute value of the Jacobian.

Triple integrals are often easier to evaluate in the cylindrical or spherical coordinates. The corresponding examples are considered on the pages