How it works ?
This applet dynamically illustrates the formation of a solid of revolution by rotating a region bounded by
y= f(x) = upper function
y= g(x) = lower function
x = a = lower limit of integration
x = b = upper limit of integration
about ANY HORIZONTAL LINE (ranging from x = -20 to x = 20 or ANY VERTICAL LINE (ranging from y = -20 to y = 20
Simply input your upper function f , your lower function g, and your lower and upper limits of integration. Choose your options, and watch what happens.
Note:
You can also change a and b values by moving the pink and blue points (respectively) on the x-axis. You can also change the axis of rotation by moving the purple point on its respective axis.
To explore in Augmented Reality, see the directions below this applet.
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device.
2) Click on the 3 horizontal bars (upper left). Select OPEN.
3) Type in the code BZWTCPfd. (It IS case sensitive).
- Note this string of characters = the last 8 digits of the URL for this resource.
- This graph defaults to rotating about a HORIZONTAL AXIS (y = some number) first.
- To rotate the area between 2 function graphs about a VERTICAL AXIS (x = some number), simply find the variable named m = false. Once you do so, change this line to m = true.
- To switch back to rotating about a HORIZONTAL AXIS, simply fine the line l = false. Change this line to l = true.
4) Once the resource loads, scroll upwards in the Algebra view (bottom) within this app. 5) The greater (higher) function is f (top most bar). You can modify this. The lesser (lower) function is h. You can modify this.
a = lower limit of integration (modifiable)
b = upper limit of integration (modifiable).
n = the angle at which you will soon rotate this region (between the 2 graphs) about the line.
c = the y-value of the horizontal line about which you will rotate (if you chose a horizontal axis).
o = the y-value of the vertical line about which you will rotate (if you chose a vertical axis).
s = the shading level of the surface (w/s = 0 being no shade and s = 1 = fully shaded).
Leave the rest of the objects alone, and you’ll be all set! Have fun exploring!