Units

# Unit 9: Triple Integrals in Cylindrical Coordinates

## The Concept

Cylindrical coordinates are a simple extension of 2D polar coordinates to 3D. Recall that, in Unit 7, the position of a point in 2D (i.e., $xy$-plane) can be described using polar coordinates $(r, \theta)$, where $r$ is the distance of the point from the origin and $\theta$ is the angle between the $x$-axis and the line segment from the origin to the point. With the addition of a third dimension, $z$-axis from the Cartesian (i.e., rectangular) coordinate system, we are able to describe a point in 3D cylindrical coordinates, i.e., $(r, \theta, z)$.

Cylindrical coordinates simply combine the polar coordinates in the $xy$-plane with the usual $z$ coordinate of Cartesian coordinates. To form the cylindrical coordinates of a point $P$, simply project it down to a point $Q$ in the $xy$-plane. Then, take the polar coordinates $(r, \theta)$ of the point $Q$. The third cylindrical coordinate is the same as the usual $z$-coordinate. It is the signed distance of point $P$ to the $xy$-plane.

## The Plot

Now, you should engage with the 3D plot below to understand triple integrals in cylindrical coordinates. Follow the steps below to apply changes to the plot and observe the effects:

1. Change the bounds on the triple integral in cylindrical coordinates where and represent the outermost bounds. $\alpha$ and $\beta$ are constants and they are the lower and upper bounds of angle $\theta$.  $r_1$ and $r_2$ are functions of $\theta$), and they are the lower and upper bounds of $r$.  $u_1$ and $u_2$ are functions of $r$ and $\theta$, and they are the lower and upper bounds of $z$.
2. You may also change the grid size of the 3D solid depicted on the screen for the function $f(x,y,z)$.

## Self-Checking Questions

Check your understanding by solving the following questions:

Evaluate the triple integrals $f(x,y,z)$ over the solid $E$.

1. $E = {(x,y,z)| x^2+y^2 \leq 9, x \geq 0, y\geq 0, 0 \leq z \leq 1}, f(x,y,z) = z$
2. $E = {(x,y,z)| 1 \leq x^2+y^2 \leq 9, y \geq 0, 0 \leq z \leq 1}, f(x,y,z) = x^2+y^2$

Use the graph to find the answers to these questions.

1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). 