There is a similar change of variables formula for triple integrals.
Let T be a transformation that maps a region S in uvw-space onto a
region R in xyz-space by means of the equations \[
x = g(u, v, w) \qquad y = h(u, v, w) \qquad z = k(u, v, w)
\] Jacobian of T is the following \(3 \times 3\)
\[
\frac{\partial(x, y, z)}{\partial(u, v, w)} = \begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} &
\frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} &
\frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} &
\frac{\partial z}{\partial w} \end{vmatrix} \tag{12}
\]
Under hypotheses similar to those in Theorem 9, we have the following
formula for triple integrals:
\[
\iiint_R f(x, y, z) dV = \iiint_S f(x(u,v,w), y(u,v,w), z(u,v,w)) \left|
\frac{\partial(x,y,z)}{\partial(u,v,w)} \right| du dv dw \tag{13}
\]
EXAMPLE 4 Use Formula 13 to derive the formula for
triple integration in spherical coordinates.
Show Answer
SOLUTION Here the change of variables is given by
\[
x = \rho\sin\phi\cos\theta \qquad y = \rho\sin\phi\sin\theta \qquad z =
\rho\cos\phi
\] \[
\frac{\partial(x, y, z)}{\partial(\rho, \theta, \phi)} = \begin{vmatrix}
\sin\phi\cos\theta & -\rho\sin\phi\sin\theta &
\rho\cos\phi\cos\theta \\ \sin\phi\sin\theta &
\rho\sin\phi\cos\theta & \rho\cos\phi\sin\theta \\ \cos\phi & 0
& -\rho\sin\phi \end{vmatrix}
\] \[
= \cos\phi \begin{vmatrix} -\rho\sin\phi\sin\theta &
\rho\cos\phi\cos\theta \\ \rho\sin\phi\cos\theta &
\rho\cos\phi\sin\theta \end{vmatrix} - \rho\sin\phi \begin{vmatrix}
\sin\phi\cos\theta & -\rho\sin\phi\sin\theta \\ \sin\phi\sin\theta
& \rho\sin\phi\cos\theta \end{vmatrix}
\] \[
= \cos\phi(-\rho^2\sin\phi\cos\phi\sin^2\theta -
\rho^2\sin\phi\cos\phi\cos^2\theta) -
\rho\sin\phi(\rho\sin^2\phi\cos^2\theta + \rho\sin^2\phi\sin^2\theta)
\] \[
= -\rho^2\sin\phi\cos^2\phi - \rho^2\sin^3\phi = -\rho^2\sin\phi
\] \(0 \le \phi \le \pi\) \(\sin\phi \ge 0\) \[
\left| \frac{\partial(x, y, z)}{\partial(\rho, \theta, \phi)} \right| =
|-\rho^2\sin\phi| = \rho^2\sin\phi
\] \[
\iiint_R f(x, y, z) dV = \iiint_S f(\rho\sin\phi\cos\theta,
\rho\sin\phi\sin\theta, \rho\cos\phi) \rho^2\sin\phi d\rho d\theta d\phi
\]