Area and Center of Gravity for a Crescent

Date: 9/5/95 at 16:16:7
From: Anonymous
Subject: area and center of gravity of crescent

Dear Dr. Math,	
	I need to calculate the area and center of gravity of a crescent. 
For example take a 10.00" dia circle center at x,y 0,0
add a 9.00" dia circle center at x,y 2,0.
The crescent is formed in 2 areas that are not overlapping.

Date: 9/22/95 at 20:52:39
From: Doctor Andrew
Subject: Re: area and center of gravity of crescent

Calculus is the best way I know of to solve this problem.  The equation for
a circle is: r^2 = (x-a)^2 + (y-b)^2, where (a,b) is the center of
the circle and r is the radius of the circle.  You can write separate 
equations for each circle in terms of x and in terms y.  Once you've got
these equations you'll need to find the x and y coordinates of where the 
circles intersect.  

If you draw a picture of the two circles and label the points where they
intersect and then shade the crescent this might help you solve the problem.
On the picture, draw lines from the intersection points perpendicular to the
x and y axes.  These will be the boundaries for your integration later.
If you already know how to find the area and center of mass of a shape
using calculus, then you can use those techniques to find the area and 
moment of one circle in a range and subtract the area and moment from another
circle at each dx and dy.  If you are integrating along the x axis, for 
instance, integrate between x-coordinates of the points of intersection and
subtract the contribution of one circle from the other at each dx.  If this 
doesn't make sense to you yet, write back and I can go a little more into how
to take the integrals.

Good luck!  Hope this isn't too late to help.

-Doctor Andrew,  The Geometry Forum