Rotating about X-AXIS: Creating Surfaces of Revolution in GGB AR
- Tim Brzezinski
Consider the function . In calculus, we often end up studying the solid of revolution formed by rotating the graph of a function about the X-AXIS. In GeoGebra's 3D Graphing Calculator, this is actually quite easy to do. The silent screencast below illustrates how easy this actually is.
How to Create a Surface of Revolution Formed by Rotating ANY Function Graph about X-AXIS
Try it yourself before moving forward! Note: You don't have to use the function illustrated above. You can use ANY FUNCTION!
However, GeoGebra's Augmented Reality app currently only allows users to plot to surfaces of the form . That is, z need to be written as a function of x and y. So let's first consider this: For the surface of revolution shown below, cross sections parallel to the yz-plane are CIRCLES whose radius is . To see this in action, move the LARGE BLUE POINT to the LEFT in the applet below. Note how these cross sections are always circles. The equation of such a circle is .
Move the LARGE BLUE POINT to the LEFT. Note cross sections are CIRCLES with radius = f(x).
Upon solving the equation above for z, we obtain and . Thus, given this, any surface of revolution formed by rotating the graph of a function about the X-AXIS can be consider to be 2 SURFACES PUT TOGETHER: z = a surface with POSITIVE OUPUTS (top half) z = a surface with NEGATIVE OUTPUTS (bottom half).
Thus, for , we obtain = blue surface shown below. = pink surface shown below.
Now let's see what this looks like in GeoGebra Augmented Reality. TRY IT and EXPLORE!
Note how this surface above resembles a vase. What other 3D solids can students model with only 2 surface functions within GeoGebra Augmented Reality?