# Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions

- Author:
- Tim Brzezinski

This applet dynamically illustrates the formation of a solid of revolution by rotating a region bounded by
to ) or to ).
Simply input your upper function , your lower function , and your

**= upper function****= lower function****= lower limit of integration****= upper limit of integration**about**ANY HORIZONTAL LINE**(ranging from**ANY VERTICAL LINE**(ranging from**lower**and**upper**limits of integration. Choose your options, and watch what happens.**Note:**You can also change*and***a***values by moving the***b****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!