Tag Archives: volume

Volumes of Revolution and Differentials

Here’s classic volume of revolution problem:

Determine the volume of the solid created when the region defined by y=x^3, the line x=2, and the x-axis is rotated about the x-axis.

Typical calculus classes present TWO ways to compute such volumes of revolution–orthogonal and parallel to the axis of rotation.  In this post, I argue that with substitutions involving differentials, you can actually get your choice of FOUR integrals to solve.  Hopefully one of them will be easy to solve.  Here’s how.

Slicing the volume ORTHOGONAL to the x-axis creates discs with radius y and thickness dx giving an original (orthogonal) volume integral \int \pi y^2 dx .

Method 1) I think the approach most familiar to most students is to substitute using y=x^3 to get

differential1

Method 2) Here’s my suggestion that I don’t see too many people use.  Consider substituting for the differential using x=y^{1/3} \longrightarrow dx=\frac{1}{3} y^{-2/3} dy  to get

differential2

Using the method of cylindrical shells, slicing the volume PARALLEL to the axis of rotation creates a series of annuli, each with radius y , thickness dy , and height (2-x) , giving an original (parallel) volume integral \int 2\pi y \cdot (2-x) dy .

Method 3) Substitute x=y^{1/3} into this to get

differential3

Method 4) Or you could use the other differential, y=x^3 \longrightarrow dy=3x^2 dx , to get

differential4

CONCLUSION:

No matter what approach you take, it’s important for students to understand that there is no single approach that will will work well for all volumes.  Increasing the number of representations available for manipulation generally should increase your chances of finding one that’s easy (or at least easier) to solve.

In the end, I just think it’s pretty that all four of these integrals are exactly equivalent even though they really don’t look much alike.

differential5

 

Screencasting some calculus

OK, no CAS here, but the power of screencasting allows one to add a voice-over to problems that are difficult to capture in the linear text that predominates Web-based writing.  The following two problems were requests from March 6-7, 2012 on the AP Calculus EDG.  My solutions are screencasts using the ShowMe Whiteboard app on my iPad.  Thanks to my former student (now technology consultant) Beth Holland for the lead to that great app.

  1. A region is defined by y=e^x, x=1, x=8, and the x-axis.  What is the volume of the solid that results when that region is revolved around the line y=4?  Click here for my solution.

As an aside, colleague John Burk (@occam98) pointed me to a cool article, Volumes of Revolution & 3D Modeling, that quite rightly identifies the most challenging part about volumes of revolutions for students:  visualization.  This is another reason why we need to make greater use of screencasting or other technologies to get past the walls of words.

  1. A curve is defined parametrically by x=t^2+1 and y=4t^3+3.  What is the length of that curve for -1\le t \le 0?  Click here for my solution.

I find that many students (and some teachers) have difficulty sorting out the different forms of arclength.  My screencast describes very briefly why I believe students should work from the single arclength formula, \int \sqrt{dx^2+dy^2}.