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[1]. 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[2]:

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.