# 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)

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)

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

(3)

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)

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