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# Volume of cylindrical hole removed from a sphere

Given a solid sphere of radius , what is the volume of material from a hole of radius through the center of the sphere.

First, the volume of a sphere of radius is given by Then, the volume of a sphere with a hole drilled in it is the volume of the solid of revolution generated by the region between and from to . Denoting this volume by we then have, Thus, the volume of the material removed from the sphere by drilling a hole in it is given by 1. Anonymous says:

How did you find the interval of integration to be negative root 3 to root 3?

• Mihajlo says:

It is the height of the equilateral triangle, because the radius of the base is r.

• AudioRebel says:

The trick is that the hole/tube does not end at the x-axis where f(x) = 0 but where f(x) = r.

From the middle inside the hole/tube it goes r units up and r units down. thus set the equation of the semi-sphere f(x) = r and solve for x. This will give you the limits of integration.

2. Ejnota Rom says:

I can’t undrrstand from where root of 3 come

3. Sue Harris says:

The third step of your simplification of V(T) has an extra π (Pi) coefficient in the second term.

• RoRi says:

Thanks! Fixed.