Evaluating Triple Integrals in Cylindrical Coordinates

Much like how many double integrals can be more easily evaluated by using polar coordinates, many triple integrals can be more easily evaluated by using cylindrical coordinates. Recall from the The Cylindrical Coordinate System page that any point $(x, y, z) \in \mathbb{R}^3$ can be represented as an ordered triple $(r, \theta, z)$. In essence, we first take the point $(x, y)$ on the $xy$ plane and write it in polar coordinates $(r, \theta)$, and then go up/down based on the value of the $z$ coordinate to get our cylindrical coordinates $(r, \theta, z)$. Furthermore, the transformation from $(x, y, z)$ to $(r, \theta, z)$ is given by the following set of equations:

(1)

\begin{align} \left\{\begin{matrix} x = r \cos \theta\\ y = r \sin \theta\\ z = z \end{matrix}\right. \end{align}

It is also important to note that $x^2 + y^2 = (r \cos \theta)^2 + (r \sin \theta )^2 = r^2(\cos ^2 \theta + \sin^2 \theta) = r^2$, and that $\frac{y}{x} = \frac{r \sin \theta}{r \cos \theta} = \tan \theta$.

Now suppose that we want to integrate a three variable function $w = f(x, y, z)$ over a region $E$ in $\mathbb{R}^3$, and suppose $E$ is a type 1 region given by $E = \{ (x, y, z) \in \mathbb{R}^3 : (x, y) \in D : u_1(x, y) ≤ z ≤ u_2(x, y) \}$. Then the triple integral of $f$ over $E$ is given by:

(2)

\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iint_D \left [ \int_{u_1(x, y)}^{u_2(x, y)} f(x, y, z) \: dz \right ] \: dA \end{align}

Furthermore, suppose that the region $D$ can be expressed as $\{ (r, \theta) : h_1(\theta) ≤ r ≤ h_2(\theta), \alpha ≤ \theta ≤ \beta \}$ where $x = r \cos \theta$ and $y = r \sin \theta$.

Then we can rewrite our triple integral in terms of cylindrical coordinates and we have that:

(3)

\begin{align} \quad \iiint_E f(x, y, z) \: dV = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} \int_{u_1(r \cos \theta, r \sin \theta)}^{u_2(r \cos \theta, r \sin \theta)} f(r \cos \theta, r \sin \theta, z) r \: dz \: dr \: d \theta \end{align}

If $E$ is a type 2 region given by $E = \{ (x, y, z) \in \mathbb{R}^3 : (y, z) \in D, g_1 (y, z) ≤ x ≤ g_2(y, z) \}$ and if $D$ can be expressed as $(r, \theta) : h_1(\theta) ≤ r ≤ h_2(\theta), \alpha ≤ \theta ≤ \beta \}$ where $y = r \cos \theta$ and $z = r \sin \theta$, then we can rewrite our triple integral in terms of cylindrical coordinates and we have that:

(4)

\begin{align} \quad \iiint_E f(x, y, z) \: dV = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} \int_{u_1(r \cos \theta, r \sin \theta)}^{u_2(r \cos \theta, r \sin \theta)} f(x, r \cos \theta, r \sin \theta) r \: dx \: dr \: d \theta \end{align}

Lastly, if $E$ is a type 3 region given by $\{ (x, y, z) \in \mathbb{R}^3 : (x, z) \in D, g_1 (x, z) ≤ y ≤ g_2(x, z) \}$ and if $D$ can be expressed as $(r, \theta) : h_1(\theta) ≤ r ≤ h_2(\theta), \alpha ≤ \theta ≤ \beta \}$ where $x = r \cos \theta$ and $z = r \sin \theta$, then we can rewrite our triple integral in terms of cylindrical coordinates and we have that:

(5)

\begin{align} \quad \iint_E f(x, y, z) \: dV = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} \int_{u_1(r \cos \theta, r \sin \theta)}^{u_2(r \cos \theta, r \sin \theta)} f(r \cos \theta, y, r \sin \theta) r \: dy \: dr \: d \theta \end{align}

All of the formulas above are especially useful when the region $E$ can be expressed nicely in terms of cylindrical coordinates. Namely, if $E$ is some region obtained from a cylinder, it is often easier to evaluate the integral of $f$ over $E$ by converting to cylindrical coordinates first.