**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

*, and your*

**g****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!