Polar Graphs

(You need Java Runtime Environment (JRE) to run GraphFunc in this website)

 

Use GraphFunc utility online to sketch the following polar graphs and find its derivatives at.  Demonstrate the computation of volume and surface area that is formed by revolving a polar graph over a given interval about the x-axis.  (See demo)

 

 

  1. , .
  2. , .
  3. , .
  4. , .
  5. (Butterfly Curve), .

 

 

Sketch Polar Graph and Find Its Derivatives

 

a., .

Begin by going to http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be loaded into your computer. (Take a few seconds)

  1. Select Polar item from Cartesian dropdown list box.
  2. Enter the function expression: 2*(1-cos(t)) at the command-line - (Note: t is theta)
  3. Click on the Graph It! button to plot the graph.
  4. Enter the valueinto the text field that has the label marked as t = . Then click on the Calc button to compute the derivatives and other values.  See Figure 1.
  5. Click on the Zoom In or Zoom Out button to adjust the graph.

 

Note:

 

 

 

Figure 1:

 

 

b., .

Begin by going to http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be loaded.

 

 

Figure 2:

 

c. , .

Begin by going to http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be loaded.

 

Figure 3:

 

d.  

Follow the instruction of (b). The result is depicted in Figure 4.

Figure 4:

 

e.  

Begin by going to http://graph.seriesmathstudy.com .

 

Figure 5:

 

(More information on this curve in Figure 5, see the articles The Butterfly Curve, American Mathematical Monthly 96, No. 5, May 1989 and A Study in Step Size, Mathematics Magazine 70, No. 2, April 1997 by Temple H. Fay.)

 

In addition, the above polar equations can be viewed in three-dimensional by selecting the View 3D item from the View 2D dropdown list box.  The Figure 6 shows the Butterfly Curve is plotted in three-dimensional polar coordinates. Notice that dragging a graph in the View 3D mode causes the motion of the graph may be slow if the interval between the lower and upper bounds is large.

 

Figure 6: - Butterfly Curve in three-dimensional polar coordinates

 

Computing The Volume and Surface Area of Solid of Revolution

 

    Use GraphFunc utility to plot the polar functionwith, and compute its volume and surface area formed by revolving the graph about the x-axis asgoes from 0 to.  Begin by going to the site http://graph.seriesmathstudy.com.  Then do the following steps:

 

  1. Select the View 3D item from View 2D dropdown list box.
  2. Select the Polar item from Cartesian dropdown list box.
  3. Enter t (use t as) into the command line.
  4. The lower and upper bounds at the labels t From and To are 0 and 2*pi.
  5. Click on the Graph It button to see its 3D graph (but not end yet).
  6. Select the x-axis item from Rotation dropdown list box forfrom 0 to(remember t is from 0 to pi for the text fields From and To).  The graph is displayed and its shape looks like a fruit apple (see the red lines). 
  7. Click on the Volume button to get the volume (approximate 25.46).
  8. Click on the Surface Area button get the surface area (approximate 42.32).

 

The result is depicted in Figure 7.

 

Figure 7: The apple-shape in red is generated by revolvingaround the x-axis for.

 

It is not difficult to verify the above result (computed volume) using the definition.  Indeed, the volume generated by revolvingfrom 0 toabout the x-axis is expressed in terms of integral, namely

       

 

 

 

       

 

 

.

 

Thus,

(= 25.4609764183109).

 

For the general,, we get.

 

 

    The next example is the Butterfly Curve, . 

 

In Figure 8 shows the Butterfly Curve is revolved about the x-axis asis from 0 to.  To create this graph forand get the computed volume and surface area for, follow the steps as described in Figure 7 above.  The limit values at the text fields From and To in this example are 0 and pi/2.  The Figure 9 displays the computation result of volume and surface area of the Butterfly Curve.

Figure 8: Butterfly Curve is revolved about the x-axis.

 

 

Figure 9: This shows the volume and surface area of the Butterfly Curve of revolution in pink for.

 

 

 

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