1. ## polar coordinates area

Find the area of the region that lies inside circle R=1 and out side the cardioid r=1-cos(x) sketch the graph

2. Originally Posted by gracy
Find the area of the region that lies inside circle R=1 and out side the cardioid r=1-cos(x) sketch the graph
Hello Gracy,

I'm not quite certain which area you mean.

I've attached a drawing of the circle and the cardioide where I've greyed the region which I believe should be calculated. Am I right?

EB

3. Hello, Gracy!

Find the area of the region that lies inside circle $\displaystyle r=1$
and outside the cardioid $\displaystyle r\:=\:1-\cos x$
Sketch the graph.
Code:
                          |
o   o   |
o        * o *
o       *     |  o::*
o       *       |    o::*
o       *        |    o:::*
|    o:::::
o       *         | -o::::::*
----o-------*---------o:::::::::*----
o       *         |  o::::::*
|    o:::::
o       *        |    o:::*
o       *       |    o::*
o       *     |  o  *
o        * o *
o   o   |
|

The curves intersect when: .$\displaystyle 1 \:= \:1 - \cos x$
. . Then: .$\displaystyle \cos x \,= \,0\quad\Rightarrow\quad x\:=\:-\frac{\pi}{2},\:\frac{\pi}{2}$

Due to the symmetry, we can integrate from $\displaystyle 0$ to $\displaystyle \frac{\pi}{2}$ and double.

The shaded area is: .(Area of the circle) - (Area of the cardiod)

Therefore: .$\displaystyle A \;=\;2 \times \frac{1}{2}\int^{\frac{\pi}{2}}_0\left[1^2 - (1 - \cos x)^2\right]\,dx$

Go for it!