Lengths and Surface
Areas of Polar Curves
Section 10.6b
Length of a Polar Curve
Start with a parametrization of r  f   ,      :
x  r cos   f   cos  y  r sin   f   sin 
   
Length of a parametric curve (from Section 10.1):
L


2
2
 dx   dy 

 
 d
 d   d 
Inside the square root:
 f  cos   f sin     f  sin   f cos  
2
2
Length of a Polar Curve
 f  cos   f sin     f  sin   f cos  
2
2
  f  cos    2 ff  cos  sin    f sin  
2
2
  f  sin    2 ff  cos  sin    f cos  
2
 f 
2
2
2
2

 cos   sin     f   cos   sin  
2
  f    f    r 
2
2
2
2
2
 dr 


 d 
2
Now substitute back into the original formula…
Length of a Polar Curve
If r  f   has a continuous first derivative for
     and if the point P  r ,  traces the curve
r  f   exactly once as  runs from  to  , then
the length of the curve is
L


2
 dr 
r 
 d
 d 
2
Length of a Polar Curve
Find the length of the given cardioid.
r  1  cos 
L


2

0
2
Check the graph for the angle interval.
2
 dr 
r 
 d
 d 
2
1  cos     sin   d
2
2

1  2 cos   cos   sin  d

2  2cos  d  8
0
2
0
2
2
(Using NINT)
Area of a Surface of Revolution
If r  f   has a continuous first derivative for     
and if the point P  r ,   traces the curve r  f   exactly
exactly once as  runs from  to  , then the areas of the
surfaces generated by revolving the curve about the x- and
y-axes are given by the following formulas:
Revolution about the x-axis  y  0  :
S


2
 dr 
2 r sin  r  
 d
 d 
2
Area of a Surface of Revolution
If r  f   has a continuous first derivative for     
and if the point P  r ,   traces the curve r  f   exactly
exactly once as  runs from  to  , then the areas of the
surfaces generated by revolving the curve about the x- and
y-axes are given by the following formulas:
Revolution about the y-axis  x  0  :
S


2
 dr 
2 r cos  r  
 d
 d 
2
Area of a Surface of Revolution
Find the area of the surface generated by revolving the righthand loop of the given lemniscate curve about the y-axis.
r  cos 2
2
Check p.565 for a graph and diagram…
r  cos 2 traces the entire graph on 

4
 

4
First, evaluate the inside of the square root:
 dr 
r 

 d 
2
2


cos 2

2
1 2
1

   cos 2   2sin 2  
2

2
Area of a Surface of Revolution
Find the area of the surface generated by revolving the righthand loop of the given lemniscate curve about the y-axis.
r  cos 2
2


Check p.565 for a graph and diagram…
1 2
1

 cos 2    cos 2   2sin 2  
2

2
2
sin 2
 sin 2 
 cos 2   
  cos 2  cos 2
cos 2 

2
1
cos 2  sin 2


cos 2
cos 2
2
2
2
Now substitute back
into the original
formula…
Area of a Surface of Revolution
Find the area of the surface generated by revolving the righthand loop of the given lemniscate curve about the y-axis.
r  cos 2
2
Check p.565 for a graph and diagram…
2
 dr 
S   2 r cos  r  
d


 d 
 4
1
  2 cos 2 cos 
d
 4
cos 2

2
 2 
 4
 4
cos  d  2 sin   4  2 2
 4