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### 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
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

Associated Topics:
College Calculus

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