AP CALCULUS BC
Section Number:
10.5
LECTURE NOTES
Topics: Area and Arc Length in Polar Coordinates
- Area
MR. RECORD
Day: 42
Area of a Polar Region
The development of a formula for the area of a polar region parallels that for the
area of the region on the rectangular coordinate system. The primary difference
is that in polar, we use sectors of circles instead of rectangles to partition off our
region.
n
Consider the function given by r  f ( ) ,
where f is continuous and nonnegative on
the interval given by      . The region
bounded by f and the radial lines    and    is what we wish to find.
If we partition  ,   into n subintervals where   0  1  2 
 n1   n  
And let the radius of the ith sector be f i  and the central angle of the ith sector be
 
n
  ,
n
2
1
Then our area can be approximated as A       f i   .
i 1  2 
Taking the limit as n   produces
AREA IN POLAR COORDINATES
1 n
2
A  lim   f (i )  
If f is continuous and nonnegative on the interval  ,   ,
x  2
i 1
0      2 , then the area of the region bounded by the graph of

2
1
r  f ( ) between the radial lines    and    is given by
   f    d
2

2
1
A    f   d
2
Example 1: Finding the Area of a Polar Region.
Find the area of one petal of the rose curve given by r  3cos3 . (See Example 4 from Section 10.4)
Example 2: Finding the Area Bounded by a Single Curve.
Find the area of the region lying between the inner and outer loops of the limaçon r  1  2sin  .
Begin by using your TI-Nspire to sketch a graph of the curve to aid in determining the boundaries of
integration. Compare your result with the sketch to the right.
After setting up the appropriate integral expressions, you may use your TI-Nspire to evaluate.
Scan this QR Code with your
smart device to see a video
explanation of this problem
Points of Intersection of Polar Graphs
ACTIVITY: 1. Try Locating the points of intersection of the polar graphs r  1 2cos and r  1 by solving
algebraically.
2. Use your TI-Nspire calculator to sketch both graphs. What do you notice concerning the
points of intersection?
Example 3: Finding the Area of a Region Between Two Curves.
Find the area of the region common to the two regions bounded by the following curves.
r  6cos 
r  2  2cos 
AP CALCULUS BC
Section Number:
10.5
LECTURE NOTES
Topics: Area and Arc Length in Polar Coordinates
- Arc Length
- Surface Area of Revolution
Arc Length in Polar Form
THEOREM 10.14
ARC LENGTH OF A POLAR CURVE
If f be a function whose derivative is continuous on the interval,      .
The length of the graph of r  f   from    to    is


s    f     f    d  
2
2


2
 dr 
r 
 d
 d 
2
Example 4: Finding the Length of a Polar Curve.
Find the length of the arc from   0 to   2 .
r  f ( )  2  2cos 
Area of a Surface of Revolution
THEOREM 10.14
AREA OF A SURFACE OF REVOLUTION
If f be a function whose derivative is continuous on the interval,      .
The area of the surface formed by revolving the graph of of r  f   from
   to    about the indicated line is

S  2  f ( )sin   f     f    d
2
2
About the polar axis


S  2  f ( ) cos   f     f    d

2
2
About the line


2
MR. RECORD
Day: 43
Example 5: Finding the Area of a Surface of Revolution.
Find the area of the surface formed by revolving the circle r  f ( )  cos  about the line  

.
2