Change of Variables in Triple Integrals Examples 1

Recall from the Change of Variables in Triple Integrals page that if we have a three variable real-valued function $w = f(x, y, z)$ that is integrable on $E$ where $E$ is a region on the $xyz$-plane and $S$ 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 while the functions $x$, $y$, and $z$ and their first partial derivatives with respect to $u$, $v$, and $w$ are continuous on the domain $S$ then:

(1)

\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}

We will now look at some examples of change of variables in triple integrals.

Example 1

Verify the formula for the change of variables using cylindrical coordinates.

Recall that for $0 ≤ r$, $0 ≤ \theta ≤ 2\pi$, and $z \in \mathbb{R}$, the change of variables for points $(x, y, z)$ to points $(r, \theta, z)$ is given by the following transformation formulas:

(2)

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

Let's compute the Jacobian $\frac{\partial (x, y, z)}{\partial (r, \theta, z)}$:

(3)

\begin{align} \quad \frac{\partial (x, y, z)}{\partial (r, \theta, z)} = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z} \\ \end{vmatrix} = \begin{vmatrix} \cos \theta & -r \sin \theta & \frac{\partial x}{\partial z} \\ \sin \theta& r \cos \theta & 0 \\ 0& 0 & 1 \\ \end{vmatrix} = \begin{vmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{vmatrix} = r \cos^2 \theta + r \sin^2 \theta = r \end{align}

Therefore $\biggr \rvert \frac{\partial (x, y, z)}{\partial (r, \theta, z)} \biggr \rvert = \mid r \mid = r$. Thus the formula for the change of variables in cylindrical coordinates is given by:

(4)

\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iiint_S f(r \cos \theta, r \sin \theta, z) r \: dz \: dr \: d \theta \end{align}