This applet dynamically illustrates the formation of a solid of revolution by rotating a region bounded by [br][br][math]y=f\left(x\right)[/math] [b]= upper function[/b][br][math]y=g\left(x\right)[/math] [b]= lower function[/b][br][math]x=a[/math] [color=#ff00ff][b]= lower limit of integration[/b][/color][br][math]x=b[/math] [color=#1e84cc][b]= upper limit of integration[/b][/color][br][br]about [b][color=#9900ff]ANY HORIZONTAL LINE[/color][/b] (ranging from [math]x=-20[/math] to [math]x=20[/math]) or [color=#9900ff][b]ANY VERTICAL LINE[/b][/color] (ranging from [math]y=-20[/math] to [math]y=20[/math]). [br][br]Simply input your upper function [math]f[/math], your lower function [math]g[/math], and your [color=#ff00ff][b]lower[/b][/color] and [b][color=#1e84cc]upper[/color][/b] limits of integration. [br]Choose your options, and watch what happens. [br][br][b]Note: [/b][br]You can also change [i][b][color=#ff00ff]a[/color][/b][/i] and [i][color=#1e84cc][b]b[/b][/color][/i] values by moving the [b][color=#ff00ff]pink[/color][/b] and [color=#1e84cc][b]blue[/b][/color] points (respectively) on the [i]x[/i]-axis. [br]You can also [b][color=#9900ff]change the axis of rotation[/color][/b] [b][color=#9900ff]by moving the purple point on its respective axis. [br][br][/color][color=#1e84cc]To explore in Augmented Reality, see the directions below this applet. [/color][/b]

1) Open up GeoGebra 3D app on your device. [br][br]2) Click on the 3 horizontal bars (upper left). Select [b]OPEN[/b]. [br][br]3) Type in the code [b]BZWTCPfd[/b]. (It IS case sensitive). [br] Note this string of characters = the last 8 digits of the URL for this resource. [br][br] This graph defaults to rotating about a HORIZONTAL AXIS (y = some number) first. [br][br] To rotate the area between 2 function graphs about a VERTICAL AXIS (x = some number), simply find the [br] variable named [b]m = false[/b]. Once you do so, change this line to [b]m = true[/b]. [br][br] To switch back to rotating about a HORIZONTAL AXIS, simply fine the line [b]l = false[/b]. [br] Change this line to [b]l = true[/b]. [br][br]4) Once the resource loads, scroll upwards in the Algebra view (bottom) within this app. [br][br]5) The greater (higher) function is f (top most bar). You can modify this. [br] The lesser (lower) function is h. You can modify this. [br] [color=#ff0000][b]a = lower limit of integration (modifiable)[/b][/color][br] [color=#1e84cc][b]b = upper limit of integration (modifiable). [/b][/color][br] [b]n[/b] = the angle at which you will soon rotate this region (between the 2 graphs) about the line. [br] [color=#9900ff][b]c = the y-value of the horizontal line about which you will rotate (if you chose a horizontal axis).[br] o = the y-value of the vertical line about which you will rotate (if you chose a vertical axis). [/b] [/color] [br] s = the shading level of the surface (w/s = 0 being no shade and s = 1 = fully shaded). [br] [br]Leave the rest of the objects alone, and you'll be all set! Have fun exploring!