Main

Volumes

Finding the volume element

Just as area was computed by finding the area element and integrating, volume is computed by determining the volume element (i.e. the volume of a slice) and then integrating:

The difficulty is in finding a suitable volume element . Once that is chosen, the rest is a matter of evaluating the integral.

Example

Compute the volume of a cylinder of radius and height using several different volume elements :

First, consider making a slice perpendicular to the base of the cylinder:

This gives a rectangular slice whose height is , the same as the cylinder. The width of the rectangle can be determined by looking at an overhead view of the cylinder. Let be the distance of the slice from the center of the cylinder (so ranges from to as the slice sweeps across the cylinder):

Doing a little algebra, we find that the width of the rectangle is . Finally, the thickness of the slice is , and so the volume element in this case is

Integrating this requires the trigonometric substitution . There are easier volume elements we could choose, as we shall see.

Another way to slice is to make cuts parallel to the base of the cylinder. Let denote the distance of the slice from the base of the cylinder:

Then each slice is a circle of radius and thickness . Thus

and ranges from 0 to , so the volume is

Another possible choice is a wedge shaped volume element. Let be the angle that the wedge forms with a fixed axis (so ranges from 0 to ):

Here, the area of the sector of the circle is . Thus the volume of the wedge is

Thus the volume is

One final option is to use cylindrical shells. Let be the radius of the shell, so that ranges from 0 to as the shells sweep through the cylinder.

The height of the cylindrical shell is and the thickness of the shell is . Recalling that the lateral surface area of a cylinder is , we have

Integrating gives that the volumes is

Example

Find the volume of a sphere of radius . First, by using discs as the volume element (shown on left below). Then use cylindrical shells as the volume element (shown on the right below). Finally, use a spherical shell for the volume element, as shown in the third diagram.

Let be the distance from the center of the disc to the center of the sphere (so ranges from to as the discs sweep across the sphere). Then drawing a right triangle shows that the radius of the disc is (since the radius of the sphere is ). See the diagram below:

The thickness of the disc is , and so the volume of the disc is (the area of the disc times its thickness), and so the volume of the sphere is

For the cylindrical shell, let be the radius of the cylinder (so ranges from 0 to as the cylinders sweep out the sphere). Then by drawing in a right triangle, one finds that the height of the cylinder is :

Recall that the lateral surface area of a cylinder with radius and height is . Thus, the lateral surface area of the cylinder is . The thickness of the shell is , and so the volume element is . It follows (after making the -substitution ) that the volume of the sphere is

Finally, for the spherical shell, let denote the radius of the spherical shell:

Recall that the surface area of a sphere of radius is . Therefore, the volume of the spherical shell (i.e. our volume element) is

Note that to sweep over the entire sphere, must range from 0 to . Therefore,

Example

Find the volume of a cone of base radius and height .

The easiest choice for volume element is a slice parallel to the base of the cone, which gives a disc. Let be the distance from the tip of the cone to the center of the disc (so ranges from 0 to as the disc sweeps across the cone), and be the radius of the disc:

The volume element is the area of the disc, , times the thickness of the disc, . It remains to find in terms of . In the cutaway in the figure on the right above, one sees that by similar triangles, , and so . Thus, the volume element is

Thus, the volume of the cone is

Example

Find the volume of a square pyramid of base edge and height .

Again, use slices parallel to the base. Let be the distance from the tip of the cone to the center of the slice (so ranges from 0 to ), and let be half of the side length of the slice.

As shown in the above cutaway, one finds by similar triangles that , and so . Therefore, the area of a slice is , and the thickness of a slice is , so the volume element is

And so the volume of the pyramid is

Example

Show that the volume of a generalized cone of base area and height is . Explain the reason for the factor of .

Let be the distance from the tip of the cone to the slice.

Because the linear dimensions of the slice grow proportionally with (e.g. the length of the slice is times the length of the base), the area of the slice will grow proportionally with the square of . This means that

Thus, the volume element is , and it follows that the volume of the cone is

The factor of comes from the fact that the volume element is proportional to the square of , hence the integral has a , which produces a factor of by the power rule.


EXERCISES

  • Find the volume of the following solid: for , the intersection of the this solid with the plane perpendicular to the x-axis is a circular disc of radius .
  • The base of a solid is given by the region lying between the y-axis, the parabola , and the line in the first quadrant. Its cross-sections perpendicular to the y-axis are equilateral triangles. Find the volume of this solid.
  • The base of a solid is given by the region lying between the y-axis, the parabola , and the line . Its cross-sections perpendicular to the y-axis are squares. Find the volume of this solid.
  • Find the volume of the solid whose base is the region enclosed by the curve and the x-axis from to and whose cross-sections perpendicular to the x-axis are semicircles.
  • Consider a cone of height over a circular base of radius . We computed the volume by slicing parallel to the base. What happens if instead we slice orthogonal to the base? What is the volume element obtained by taking a wedge at angle of thickness ?
  • Consider the following solid, defined in terms of polar coordinates: ; ; . Can you describe this shape? Compute its volume.
  • Consider the following solid, defined in terms of polar coordinates: ; ; . Can you describe this shape? Compute its volume.
  • Challenge: compute the volume intersection of the (infinite) cylinders of radius centered along the x and y axes in 3-d. That is, compute the volume of intersection of

in the 3-dimensional space.