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.