14.2 Evaluation of triple integrals in cylindrical coordinates

Integration Setup

Setting Up Triple Integrals and Integration Limits

A triple integral evaluates a function $f(x, y, z)$ over a three-dimensional region $E$, written as $\iiint_E f(x, y, z) \, dV$, where $dV$ is the volume element.

To set one up, you need to determine the integration limits that describe the boundaries of $E$. These limits come from the geometry of the region (a rectangular box, cylinder, cone, sphere, etc.) and often depend on the other variables.

The order of integration specifies which variable you integrate first, second, and third. Common orders include $dz \, dy \, dx$, $dz \, dr \, d\theta$, and so on. Choosing the right order can make a problem dramatically simpler. Pick the order that gives you the cleanest limits.

Iterated Integrals and Evaluation

Iterated integrals break the triple integral into three nested single integrals. Here's how you evaluate one:

1. Start with the innermost integral. Treat all other variables as constants and integrate with respect to the innermost variable.
2. Substitute that result into the middle integral. Now integrate with respect to the middle variable, still treating the outermost variable as a constant.
3. Evaluate the outermost integral to get your final answer.

For example, with the integral $\int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx$:

• First compute $\int_e^f f(x, y, z) \, dz$, holding $x$ and $y$ constant.
• Plug that result into $\int_c^d [\cdots] \, dy$ and evaluate, holding $x$ constant.
• Finally, evaluate $\int_a^b [\cdots] \, dx$.

The key thing to watch: when limits depend on other variables (e.g., $z$ goes from $0$ to $\sqrt{r^2 - x^2}$), you must respect that dependency at each stage.

Cylindrical Coordinates

Cylindrical Volume Element and Change of Variables

Cylindrical coordinates $(r, \theta, z)$ are the natural choice whenever the region has symmetry about the $z$-axis. The three coordinates are:

• $r$: distance from the $z$-axis (always $\geq 0$)
• $\theta$: angle measured in the $xy$-plane from the positive $x$-axis
• $z$: height, same as in Cartesian coordinates

The conversion formulas from cylindrical to Cartesian are:

• $x = r\cos\theta$
• $y = r\sin\theta$
• $z = z$

When you switch to cylindrical coordinates, the volume element changes:

$dV = r \, dr \, d\theta \, dz$

That extra factor of $r$ is critical. It comes from the Jacobian determinant $\left|\frac{\partial(x,y,z)}{\partial(r,\theta,z)}\right| = r$. Forgetting this $r$ is one of the most common mistakes on exams. Every integrand $f(x,y,z)\,dV$ becomes $f(r\cos\theta,\, r\sin\theta,\, z)\; r\, dr\, d\theta\, dz$.

Setting up limits in cylindrical coordinates typically follows this pattern:

1. Determine the range of $\theta$ (often $0$ to $2\pi$ for full revolution, or a smaller interval for a wedge).
2. Determine the range of $r$ (often $0$ to some function of $\theta$, or just a constant for a full cylinder).
3. Determine the range of $z$ (often between two surfaces, expressed in terms of $r$ and $\theta$).

Cylindrical Shells and Integration

The cylindrical shell method connects single-variable techniques to the triple integral framework. A cylindrical shell is a thin hollow cylinder at radius $r$ with infinitesimal thickness $dr$.

For a solid of revolution, the volume using shells is:

$V = \int_a^b 2\pi r \, h(r) \, dr$

where $h(r)$ is the height of the shell at radius $r$, and $r$ ranges from $a$ to $b$.

Example: Volume of a cone with base radius $R$ and height $H$.

The cone's surface slopes linearly, so at radius $r$, the height is $h(r) = \frac{H}{R}(R - r)$. Setting up the integral:

$V = \int_0^R 2\pi r \cdot \frac{H}{R}(R - r)\, dr = \frac{2\pi H}{R}\int_0^R (Rr - r^2)\, dr$

Evaluating:

$= \frac{2\pi H}{R}\left[\frac{Rr^2}{2} - \frac{r^3}{3}\right]_0^R = \frac{2\pi H}{R}\left(\frac{R^3}{2} - \frac{R^3}{3}\right) = \frac{2\pi H}{R}\cdot\frac{R^3}{6} = \frac{1}{3}\pi R^2 H$

This matches the well-known cone volume formula, which is a good sanity check. Whenever you can verify your triple integral result against a known formula, do it.