Going in Circles?

First things first. For the first circle, A

r^2 = a^2 + 16

For the next two circles,

r1 = a1

r2 = a2

There is no intersection between circles A1 and A2 since they are both tangential to the y axis and in different sides of it.

Circle A intersects the x axis at -a +/- sqrt(16+a^2)

The positive intersection 0 < -a + sqrt(16+a^2) < 4

Circle A1 intersects the x axis at 2*r1

If r1 > r2, then circle A2 is completely inside circle A, and part of circle A1 is inside A.

We can find the points of intersection between A and A2 by setting their equations equal to each other. They will intersect at

x = [ r*r2 + r1*r2 + r1^2 - a^2 ] / [2 *(a + r1) ]

To find the area of circle A1 that is outside of A, we can integrate 2*(y1 - y)*dx from

x = [ r*r2 + r1*r2 + r1^2 - a^2 ] / [2 *(a + r1) ]

to x = -a + sqrt(16+a^2)

Then integrate 2*y1*dx from

x = -a + sqrt(16+a^2)

to x = 2*r1

There will be a bunch of cumbersome algebra. Good luck with it.

**EDIT**

If all of A1 and A2 are inside A, then that means

a = a1 - a2

because the smaller circles will be tangential to the large one.

Other equations:

2a1 = r - a = sqrt(a^2 + 16) - a

2a2 = r + a = sqrt(a^2 + 16) + a

So r^2 - a1^2 - a2^2

= a^2 + 16 - 1/4 * (4a^2 + 32)

= 8

A - A1 - A2 = 8π