POLAR COORDINATES
299
35.75
Find the area between the inner and outer ovals of r
2 = 1 + sin 0 in the first quadrant (see Fig. 35-18).
The area inside the outer oval is
The area inside the inner oval is the same as
Hence
35.76
Find the arc length of the spiral r = 6 from 0 = 0 to 0 = 1.
The general arc length formula is
In this case, drldO = 1, and we have L =
Letting 8 — tan u, dO = sec
2 u du. we obtain
35.77
Find the arc length of the spiral r = e" from 0 = 0 to 0 = In 2.
35.78
Find the length of the cardioid r = l-cos0 (Problem 35.34).
So,
35.79
Find the length of r=6
2
from 6=0 to 0 = VS.
35.80
Find the length of
from 0 = 0 to 0 = 2-77-.
Then
35.81
Find the arc length of r = 116 from 0 = 1 to
Let
We
Then
get
Fie. 35-18
Hence,
sec
3 u du = 5 (sec u tan w + In |sec u +