Wednesday, January 13, 2010

calculus - volumes by revolution

Please Note: This file marks the first time I have posted a GSP5 file, which will not work in earlier versions of the program. GSP5 allows you to save a file to be GSP4-compatible (provided the sketch does not make use of new features) and I will try to do this when it is possible.

This sketch may be a handy tool for depicting the volume that is generated by revolving an area about a vertical or horizontal axis. The volume is shown as a wire-frame "family of curves", a locus of a locus, and one of several fantastic new capabilities in GSP 5. This sketch will not work in GSP 4.

The volume can be tilted at any angle, and the movement through a small range of angles will allow one to observe the shape of the volume. The vertical and horizontal axes can each be moved. The area can be 'swept' around the axis.

The default functions define an area that can be integrated with respect to x, but not (explicitly)with respect to y. Therefore, one axis forces us to use cylindrical shells while the other direction of revolution will allow only discs. Discs and cylindrical shells can be shown for each individual value of x, as determined by a point moving through the interval that defines the area.
I have now updated the file by adding another page. The new sketch shows an example which could be integrated by shells with respect to x or by discs with respect to y. I hope that the sketch will work nicely as an illustration. The functions in the new page of the sketch can be changed, but the example is not general but rather discusses a case in which both functions are one-to-one and can therefore be negotiated with either variable.

Download GSP5 file:

Keywords: calculus, volumes, integration, revolution, GSP 5, family of functions

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