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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