Cylindrical Coordinates

Many physical and geometric problems involve functions that depend on the distance of the point (x, y, z) from the z-axis. Consider, for example, a cone like the cone shown at the left. We can describe a cone whose base radius is R and whose height is H using Cartesian coordinates by --

But it is much clearer if we use the variable r for the distance of the point (x, y, z) from the z-axis and write

This is one-half of the idea behind cylindrical coordinates. We describe a point using the coordinates

The movie at the right shows the points represented by constant values of the first coordinate as it varies from zero to one.

The movie at the right shows the points represented by constant values of the second coordinate as it varies from zero to 2 pi.

It is important to compare the units that are used in Cartesian coordinates with the units that are used in cylindrical coordinates. In Cartesian coordinates, (x, y, z), all three coordinates measure length and, thus, are in units of length. In cylindrical coordinates, (r, theta, z), two of the coordinates -- r and z -- measure length and, thus, are in units of length but the coordinate theta measures angles and is in "units" of radians. The most important part of the preceding sentence is the quotation marks around the word "units" -- radians are a dimensionless quantity -- that is, they do not have associated units.

The formulas below enable us to convert from cylindrical coordinates to Cartesian coordinates.

The formulas below enable us to convert from Cartesian coordinates to cylindrical coordinates.

1. Find the cylindrical coordinates of the point whose Cartesian coordinates are
   (1, 2, 3)

   answer

2. Find the Cartesian coordinates of the point whose cylindrical coordinates are
   (2, Pi/4, 3)

   answer

3. Describe the top and bottom of a sphere of radius R whose center is at the origin two different ways -- first, by using Cartesian coordinates and then by using cylindrical coordinates.

4. Describe the bowl z = x² + y² using cylindrical coordinates.

In the last module we found the volume of a solid of the form

Example

Find the volume of a cone whose base is a disk of radius 1.5 centered at the origin and whose height is 3.

We can describe this cone in cylindrical coordinates by

When we worked in Cartesian coordinates we broke the region D into squares and then estimated the volume over each square by looking at a prism whose height was determined by evaluating the function f at a point in the square. We use the same basic idea now except that we break the region D into sections like the sections shown in the figure at the right. These sections are formed as follows. As usual we begin with a positive integer n that determines how many sections we use and the size of the sections. The larger n is the better our estimate will be. Next we draw n rays emanating from the origin. For each i = 1, 2, ... n we draw a ray whose angle with the positive x-axis is 2 pi i / n. Next we draw concentric circles of radius j / n for j = 1, 2, 3, ...

This breaks the region D up into sections as shown in the figure above. In each of these sections we choose the midpoint as a "sample point." The polar coordinates of these midpoints are

Using a little geometry we see that the area of the section whose midpoint is s_ij is

Use your CAS window to estimate the volume of our cone in this way. You can compute the exact answer by looking at the cone obtained by rotating the line

around the x-axis. This cone has the same height and base radius as our cone. Compare your estimates with the exact answer.

The exact volume is the limit of these estimates. We define the multiple integral in cylindrical coordinates to be the limit of these estimates.

Units

We can gain some insight into the integration formulas shown at the right by looking units. In all four formulas the values of the function f(r, theta) are in units of length. In formulas (3) and (4) the variable r and the symbol dr are in units of length but the symbol d theta is dimensionless because it measures angles. Thus, the integrand in formulas (3) and (4) is in units of volume, as expected. Formulas (1) and (2) are more difficult because the symbol dA is used differently in the two formulas. In formula (2) this symbol refers to a little section of the region shown below.

In this picture units along the "x-axis" are dimensionless and units along the "y-axis" are length, so "area" is in units of length. Thus, in equation (2) the symbol dA is measured in units of length. Notice that the integrand in this equation is in units of volume as expected. In equation (1) the symbol dA refers to small sections of the region shown in the picture at the right. In this region "area" is real area and is measured in units of area or length2. Notice that in this formula the integrand in measured in units of volume as expected.

The method we developed in this example is very general and can be used to find the volume of any solid described in cylindrical coordinates by

1. Find the exact volume of our cone by evaluating the iterated integrals below.

   

2. Find the volume of the portion of a cone described by

   

3. Find the volume of a sphere of radius R.

4. Evaluate the integral

   

   where D is the disk of radius 1 centered at the origin.

5. Evaluate the integral

   

   where D is the disk of radius R centered at the origin.

6. Evaluate the integral