Polar Area and Acrlength


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 

  =  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

         A  =         r2(q)dq           

Adding up all the pieces, we arrive at


Find the area enclosed by the curve 

        r  =  2 cos q




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


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

we have

We also have that the surface area of revolution is



Find the length of the 8 petalled flower 

        r = cos(4q)


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.


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)



Use your graphing calculator to find the area that results when 

        r  =  1 + cosq 

is revolved around the y-axis.


We have 

        r'  =  -sin q

Putting this into the formula gives


The Calculator gives and answer of 6.4.


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