Polar Graphs

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

 

Demonstrate using the GraphFunc utility online to sketch the polar graphs and find its derivatives at.  Then compute the volume and surface area formed by revolving the graph of the polar equation over a given interval about the x-axis.  (Click here to see the basic demo how GraphFunc utility works.)

 

 

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

 

 

Sketch Polar Graph and Find Its Derivatives

 

a., .

Get into the website at http://graph.seriesmathstudy.com or click on this link, GraphFunc, to get it in a separate window. (You may have to wait for the applet to be loaded if the first time you load this page.)

1. Select Polar item from Cartesian dropdown list box.
2. Enter the polar 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 value pi/4 into the text field that has the label marked 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:

• Use the second Calc button when x is given.  For example, user enters -2 in the text field that has label marked x =, and clicks on the Calc button to get its computed results.  The computed results are also displayed if the user clicks with the mouse anywhere on the graph.

 

• To find the area or the arc length of this curve over a certain interval, enter the values of the lower and upper limits into the text fields that have the labels marked From and To. Then click on the Find Area button or Find Length button to get its results.

 

 

Figure 1:

 

 

b., .

Get into the website at http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be loaded.

• Select Polar item from Cartesian dropdown list box.
• Enter the polar function with this syntax: 2*cos(3*t) - (Note: t is theta)
• Enter the values of the lower and upper limits -pi and pi into the text fields that have labels marked t From and To, respectively.
• Click on the Graph It! button to plot the graph.
• If need, click on the Zoom In button or drag the left screen to an appropriate position so that the graph is viewable.
• Enter the value pi/4 in the text field (at the label t = ) and click on the Calc button to compute its derivatives and other values. See Figure 2.

 

 

Figure 2:

 

c. , .

Get into the website at http://graph.seriesmathstudy.com.

• Select Polar item from Cartesian dropdown list box.
• Enter the polar function with this syntax: 6/(6  5*sin(t)) - (Note: t is theta)
• Enter the limit values 0 and 2*pi at the labels marked as t From and To, respectively.
• Click on the Graph It! button to plot the graph.
• Click on the Zoom In button or drag the left screen to an appropriate position so that the graph is viewable.
• Enter the value pi/4 in the text field (at the label t = ) and click on the Calc button to compute its derivatives and other values. See Figure 3.

 

Figure 3:

 

d.  

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

Figure 4:

 

e.  

Get into the website at http://graph.seriesmathstudy.com .

• Select Polar item from Cartesian dropdown list box.
• Enter the polar function with this syntax: e^(cos(t)) - 2*cos(4*t) + sin(t/12)^5 - (Note: t is theta)
• Enter the values of the lower and upper bounds 0 and 8*pi at the labels marked as t From and To, respectively.
• Click on the Graph It! button to plot the graph.
• Enter the value pi/4 in the text field (at the label t = ) and click on the Calc button to compute its derivatives and other values. If the computed values from the left screen are not fully shown at the bottom of the graph, user needs to drag the graph to the upper left corner of the screen before clicking on the Calc button.
• If need, click on the Zoom In button to adjust the graph viewable. See Figure 5.

 

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, user can view the above polar equations in three-dimensional by selecting the View 3D item from the View 2D dropdown list box.  The Figure 6 depicts the graph of the Butterfly Curve in three-dimensional polar coordinates.

 

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

 

Computing The Volume and Surface Area of Solid of Revolution

 

    Use GraphFunc utility to plot a polar function and compute its volume and surface formed by revolving the graph about the x-axis.  For a simplest one such,, the solid is generated by revolving the graph ofabout the x-axis as depicted in Figure 7. To create this graph forand get the computed volume and surface area asgoes from 0 to, follow the steps from 1 to 8 shown in Figure 7.

 

1. Select the View 3D item from View 2D dropdown list box.
2. Select the Polar item from Cartesian dropdown list box.
3. Enter the polar express 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 get its 3D graph displayed.
6. Select the x-axis item from Rotation dropdown list box, and fill the valuesfrom 0 to.  (Remember t is from 0 to pi for the text fields From and To). 
7. Click on the Volume button to get the volume computed (25.46…).  The graph is displayed in the red lines.
8. Click on the Surface Area button get the surface area computed (42.32…).

 

 

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 one is we consider the Butterfly Curve, . 

 

In Figure 8 shows the Butterfly Curve is revolved about the x-axis asis from 0 to(pi).  To create this graph forand get its computed volume and surface area for, follow the steps depicted in Figure 7.  Remember to enter the values 0 and pi/2 for the text fields From and To.  The computed result is shown in Figure 9.

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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