Polar Area and Acrlength

Area

If we have a region defined by 

        r  =  r(q),      q = a   and   q = b 

what is the area of the region?  If r is the arc of a circle then we want to find the area of the sector of the circle.  If 

        q
  =  b - a 

then the area

        A  =  1/2 q r2

This is true since the area of the entire circle is p r2.  We can set up the relationship:

             A               q
                      =                          
           p r2             2p 

so that 

                    p r2 q            1
         A  =                  =          q r2             
                      2p               2


Cutting the region into tiny dq pieces, we have

                    1
         A  =         r2(q)dq           
                    2  


Adding up all the pieces, we arrive at




Exercise

Find the area enclosed by the curve 

        r  =  2 cos q

 


 

Arclength

Since the arclength of a parameterized curve is given by

       

we have that for polar coordinates, letting 

        x(q) = r(q) cos q  =  r cos q

and 

        y(q)  =  r(q) sin q  =  r sin q

we have


We also have that the surface area of revolution is

       



Example

Find the length of the 8 petalled flower 

        r = cos(4q)

Solution

We find the length of one of the petals and multiply by 8.  We see that the right petal goes through the origin at -p/8 and next at p/8.  Hence we integrate

       

This is best done with a calculator which  gives an answer of 2.26.

Exercise

Find the length of the curve 

        r = 5(1 + cos(q))     between 0 and 2p.



Surface Area of Revolution

We have the following two formulas:  If r = r(q) is revolved around the polar axis (x-axis) then the Surface area is

Surface Area (Revolved Around the x-axis)

         

 

If it is revolved around the y-axis then the resulting surface area is

 
Surface Area (Revolved Around the y-axis)

         




Example:
 

Use your graphing calculator to find the area that results when 

        r  =  1 + cosq 

is revolved around the y-axis.


Solution

We have 

        r'  =  -sin q

Putting this into the formula gives

       

The Calculator gives and answer of 6.4.

 



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