Rotating about X-AXIS: Creating Surfaces of Revolution in GGB AR

Consider the function [math]y=g\left(x\right)[/math]. [br][br]In calculus, we often end up studying the solid of revolution formed by rotating the graph of a function [math]y=g\left(x\right)[/math] about the X-AXIS. [br][br]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 [br][math]z=[/math]. That is, [i]z[/i] need to be written as a function of [i]x[/i] and [i]y[/i]. [br][br][b]So let's first consider this: [/b][br][br]For the surface of revolution shown below, [b]cross sections[/b] parallel to the yz-plane are [b]CIRCLES whose radius is [/b][math]y=f\left(x\right)[/math][b]. [/b] To see this in action, move the [b][color=#1e84cc]LARGE BLUE POINT[/color][/b] to the LEFT in the applet below. [br]Note how these cross sections are always circles. [br][br]The equation of such a circle is[br][math]y^2+z^2=\left(f\left(x\right)\right)^2[/math].
Move the LARGE BLUE POINT to the LEFT. Note cross sections are CIRCLES with radius = f(x).
Upon solving the equation above for [i]z[/i], we obtain [math]z=\sqrt{\left(f\left(x\right)\right)^2-y^2}[/math] and [math]z=-\sqrt{\left(f\left(x\right)\right)^2-y^2}[/math] . [br][br]Thus, given this, any surface of revolution formed by rotating the graph of a function [math]y=g\left(x\right)[/math] about the X-AXIS can be consider to be 2 SURFACES PUT TOGETHER: [br][br][b][color=#1e84cc]z = a surface with POSITIVE OUPUTS (top half)[/color][/b][br][color=#ff00ff][b]z = a surface with NEGATIVE OUTPUTS (bottom half). [br][/b][/color][justify][color=#ff00ff][b][br][/b][/color]Thus, for [math]f\left(x\right)=\sin\left(x\right)+3[/math], we obtain[br] [math]z=\sqrt{\left(\sin\left(x\right)+3\right)^2-y^2}[/math][b][color=#1e84cc]= blue surface shown below. [br][/color][/b][math]z=-\sqrt{\left(\sin\left(x\right)+3\right)^2-y^2}[/math][color=#ff00ff] [b]= pink surface shown below. [/b][/color][/justify]
Now let's see what this looks like in GeoGebra Augmented Reality. TRY IT and EXPLORE!
Note how this surface above resembles a vase. [br][br][b][color=#0000ff]What other 3D solids can students model with only 2 surface functions within GeoGebra Augmented Reality? [/color][/b]
– Tim Brzezinski

Information: Rotating about X-AXIS: Creating Surfaces of Revolution in GGB AR