Change of Variables in Triple Integrals

Change of Variables in Triple Integrals

Sometimes evaluating a triple integral can be difficult. Recall that if we have a two variable real-valued function, $z = f(x, y)$ that is integrable over $D$ as a region in the $xy$-plane and $S$ is a region in the $uv$-plane and $x = x(u, v)$, $y = y(u,v)$ define a one-to-one transformation and whose first partial derivatives are continuous on $S$, then we can evaluate the double integral of $f$ over $D$ by change of variables and:

(1)
\begin{align} \quad \iint_D f(x, y) \: dA = \iint_S f(x(u, v), y(u,v)) \biggr \rvert \frac{\partial (x, y)}{\partial (u, v)} \biggr \rvert \: du \: dv \end{align}

We can extend this idea further to triple integrals. Suppose that we have a three variable function $w = f(x, y, z)$ that is integrable over $E$ as a region in the $xyz$-plane and $S$ is a region in the $uvw$-plane. Also suppose that we have a one-to-one transformation defined by the equations:

(2)
\begin{align} \quad x = x(u, v, w) \\ \quad y = y(u, v, w) \\ \quad z = z(u, v, w) \end{align}

Then we can evaluate the triple integral of $f$ over $E$ by change of variables and:

(3)
\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iiint_S f(x(u, v, w), y(u, v, w), z(u,v,w)) \biggr \rvert \frac{\partial (x, y, z)}{\partial (u, v, w)} \biggr \rvert \: du \: dv \: dw \end{align}

In some cases, a change of variables can make evaluating a triple integral much simpler. We have already see examples of changing variables in triple integrals on the Evaluating Triple Integrals in Cylindrical Coordinates and Evaluating Triple Integrals in Spherical Coordinates pages.

When we evaluated triple integrals in cylindrical coordinates, for $0 ≤ r$, $0 ≤ \theta ≤ 2\pi$ and $z \in \mathbb{R}$, we used the following transformation in our change of variables:

(4)
\begin{align} \quad x = r \cos \theta \\ \quad y = r \sin \theta \\ \quad z = z \end{align}

Similarly, when we evaluated triple integrals in spherical coordinates, for $0 ≤ \rho$, $0 ≤ \theta ≤ 2\pi$ and $0 ≤ \phi ≤ \phi$, we used the following transformation in our change of variables:

(5)
\begin{align} \quad x = \rho \sin \phi \cos \theta \\ \quad y = \rho \sin \phi \sin \theta \\ \quad z = \rho \cos \phi \end{align}

We summarize the change of variables in triple integrals in the following theorem.

 Theorem 1: Let $E$ be a region on the $xyz$-plane and let $S$ be a region on the $uvw$-plane, and let the equations $x = x(u, v, w)$, $y = y(u, v,w)$, and $z = z(u, v, w)$ define a one-to-one transformation. Suppose that the functions $x$, $y$, and $z$ and their first partial derivatives with respect to $u$, $v$, and $w$ are continuous on the domain $S$ and that $f(x, y, z)$ is integrable on $E$. Then $\iiint_E f(x, y, z) \: dV = \iiint_S f(x(u, v, w), y(u, v, w), z(u, v,w)) \biggr \rvert \frac{\partial (x, y, z)}{\partial (u, v, w)} \biggr \rvert \: du \: dv \: dw$.