G.2.a)

Here's a region plotted in the -plane.

Go with
       
and calculate
       
by the method of least labor on your part.

G.2.b)

Here's another region plotted in the -plane.

Fancy dudes say that this is the cardiod described in polar coordinates
by the polar equation .

Go with
       
and calculate
       
by the method of least labor on your part.

Next, go with
       
and calculate
       
by the method of least labor on your part.

Compare the values of
        and .
Discuss how you think that the shape and position of account for the relationship between these two numbers.

G.2.c)

Here is another region plotted in the -plane.

Go with
       
        and
       
and calculate
       ,
       , and
       
by the method of least labor on your part.
Look at the formulas for and and then look at the shape and position of and explain why it was no surprise that
        and
came out the way they did.

G.2.d)

Here is another region plotted in the -plane.

Go with
       
and calculate
       
by the method of least labor on your part.

G.2.e)

Given that the region consists of everything inside the ellipse
       ,
go with
       
and calculate
       
by the method of least labor on your part.