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## Homework Statement

Find the volume of the solid that lies in between both of the spheres:

x

^{2}+y

^{2}+z

^{2}+4x+2y+4z+5=0

and

x

^{2}+y

^{2}+z

^{2}=4

## Homework Equations

This is the first chapter of the calculus III material so no double or triple integrals are needed to solve this problem.

## The Attempt at a Solution

I completed the square on the first equation and obtained:

(x+2)

^{2}+(y-2)

^{2}+(z+2)

^{2}=4

So I just have 2 intersecting spheres. Finding the volume of a "slice" of a sphere can be done using a solid of revolution, however, I would need know how far one sphere is intersecting into the other in order to calculate the volume.

My idea is to minimize the distance between (x+2)

^{2}+(y-2)

^{2}+(z+2)

^{2}=4 and the point (0,0,0). Then I will be able to figure out how "thick" the 2 "slices" of the spheres are.

However, I'm not quite sure how to do this. I know how to find the distance between points in space but solving for each variable I will wind up with 2 solutions because of the radical. Is there an easier way to do this that I am missing?

Thanks.

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