Volumes

Volumes

Many beginning calculus students struggle with Volumes, try to memorize a formula for each of the many cases (dx versus dy, horizontal versus vertical axis, etc), and wonder why often still find themselves confused.

The following is an (imperfect) transcription of my lecture notes outlining the "geometric way" of doing volumes which I use. It has received many positive comments from students as being clearer than the textbook(s)' explanation -- including one student who had taken half the course before and found this presentation much clearer -- HB.

Step I: Read the problem carefully and identify the axis of rotation and draw it. Then draw and shade the region that is being rorated about the axis.

A horizontal axis can be either the x-axis or a line y=constant -- make sure you understand why -- as in the following two examples:

A vertical axis can be either the y-axis or a line x=constant -- make sure you understand why -- as is depicted in the following two examples:

Step II: Draw a representative slice:

If you are using Discs or Washers, draw the slice perpendicular to the axis.
If you are using Shells, draw the slice parallel to the axis.

Step III: Realize what variable you are integrating with respect to:

If the slices are vertical: then you are integrating with respect
to x; the "infinitesimal width" of the slice is "dx"

If the slices are horizotal: then you are integrating with respect
to y; the "infinitesimal height" of the slice is "dy"

Now set up either:

x=b
	( ... ) dx
x=a

Or:

y=b
	( ... ) dy
y=a

...depending on whether you are integrating with respect to x or y.

Step IV: Write down or recall the Geometric Formulas which you have memorized (once you understand the pictures, "memorizing" them will take little effort, unlike trying to memorize-by-rote many different cases for "dx" and for "dy", for "vertical" versus "horizontal" etc -- there are only three geometric formulas in total). These formulas are:

Discs:

b
	(radius)² d(width)
a

Washers:

b
	[(Big Outer Radius)² - (small inner radius)²] d(width)
a

Shells:

b
	2(radius)(height) d(width)
a

NOTE: We sometimes abbreviate the integrants as "r²d(w)" (for Discs), as
"[R² - r²] d(w)" (for Washers), and "2(r)(h)d(w)" (for Shells).

Step V: Transform all the parts of your geomestric formula, writing them in terms of the variable with respect to which you are integrating. In particular replace "a" and "b", the "radius" (as well as the larger outer"Radius" and the "height" if present in your formula), and the "d(width)", all in terms of x's or else all in terms of y's.

Obviously, "d(width)" is replaced by "dx" if you are integrating with respect to x, and by "dy" if you are integrating with respect to y. To know what to replace "a" and "b" by, you must look at your region which you have sketched, based on the description in the problem. Many regions can be described both in terms of functions of x and alternatively in terms of functions of y. You will need to replace "a" and "b" either all by functions of x (possibly including constant functions) or else all by functions of y (again you might need to use the constant function of y, which looks like "a number" but is actually a function).

To know what to replace "radius" (and, if present, the larger outer "Radius", or if present, the "height") by, you will need to look at the picture you have drawn, which again depend on the volume problem you were given. The pictures below illustrate the various cases; for simplicity, the x-axis is used as the generic horizontal axis and the y-axis is used as the generic vertical axis; with slight modifications you can draw analogous pictures where the axis is, "y=3" or even "x=-1".

In the case of Discs we have:

In the case of Washers we have:

In the case of Shells we have:

If the slices are vertical:		then you are integrating with respect
to x; the "infinitesimal width" of the slice is "dx"

If the slices are horizotal:		then you are integrating with respect
to y; the "infinitesimal height" of the slice is "dy"