Volume

Skip to multiple and iterated integrals

We have seen several examples of problems that can be solved in simple situations by multiplication and in more complicated situations by integration. You may want to review the module on Units and Integration. In this module we look at another problem like this -- the volume of a simple solid like the solid at the right is the area of its base multiplied by its height. Simple multiplication works because the height is the same at every point.

For more complicated solids, like the cone shown at the right, in which the height varies from point to point simple multiplication is replaced by multiple integration. Our first problem is describing the base of a three-dimensional figure -- for example, the base of the cone shown at the right is a disk. We begin by looking at this example.

Example

Consider the set, D, of all points in R² whose distance from the origin is less than or equal to 2. We can describe this set in three different ways.

The movie at the left illustrates the second (on the left) and third (on the right) descriptions. The second description begins by looking at the possible x-coordinates of points in the disk --- x must be between -2 and +2. Then, for each x value, it looks at the range of possible y values. The third description begins by looking at the possible y-coordinates of points in the disk --- y must be between -2 and +2. Then, for each y value, it looks at the range of possible x values.

Example

Next we consider the filled in triangle, T, shown in the figure below.

The three edges of this triangle can be described as follows. The bottom edge (red): 1 <= x <= 8, y = 2. The right edge (green): x = 8, 2 <= y <= 9. The upper left edge (blue): 1 <= x <= 8, y = x + 1 or x = y - 1, 2 <= y <= 9.

This leads to two descriptions of the triangle T.

Example

Next consider the filled in triangle, R, shown in the figure below.

The three edges of this triangle can be described as follows: The bottom edge (red): 2 <= x <= 7, y = 2. The right edge (green): 4 <= x <= 7, y = 16 - 2 x or x = (16 - y)/2, 2 <= y <= 8. The left edge (blue): 2 <= x <= 4, y = 3 x - 4 or x = (y + 4)/3, 2 <= y <= 8.

This leads to two descriptions of the triangle R.

and

For the second description we break the original triangle into two pieces -- colored red and blue in the figure below.

1. Describe the set of all points in the disk of radius 3 centered at the point (2, 4) three different ways.

   answer

2. Describe the set of all points in the elliptical disk

   

   in two additional ways.

3. Describe the filled in triangle whose vertices are at the point (2, 3), (5, 5), and (7, 4) two different ways.

   answer

4. Describe the set of all points in the disk of radius 4 centered at the origin that are above the diagonal line y = x two different ways.

   answer

5. The three lines
   

   divide the plane into a number of different of pieces, one of which is a triangle. Describe the triangular piece two different ways.

   answer

In this module we will work with reasonably nice regions in R² -- regions that can be described in one of two ways.

In these situations we are often working with units of length and area. The variables x and y are usually expressed in units of length. For example, if we are working with meters then the top of the triangle at the left should be described by y = 10 meters - (2/3) x Since x is measured in meters, y is also measured in meters. The area of this triangle wil be measured in square meters.

Occasionally, we work with more complicated regions like the region shown on the left side of the picture below. This region must broken into four smaller regions each of which can be described in one of the two ways above.