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Volume of cylindrical hole removed from a sphere

Given a solid sphere of radius 2r, what is the volume of material from a hole of radius r through the center of the sphere.


First, the volume of a sphere of radius 2r is given by

    \[ V_S = \frac{4}{3} \pi (2r)^3 = \frac{32}{3} \pi r^3. \]

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 f(x) = \sqrt{4r^2 - x^2} and g(x) = r from -\sqrt{3} r to \sqrt{3} r. Denoting this volume by V_T we then have,

    \begin{align*}  V_T &= \pi \int_{-\sqrt{3} r}^{\sqrt{3} r} (4r^2 - x^2 - r^2) \, dx \\      &= \pi \left( \int_{-\sqrt{3} r}^{\sqrt{3}r} (3r^2 - x^2) \, dx \right) \\      &= 6 \sqrt{3} \pi r^3 - 2 \sqrt{3} \pi r^3 \\      &= 4 \sqrt{3} \pi r^3. \end{align*}

Thus, the volume of the material removed from the sphere by drilling a hole in it is given by

    \[ \frac{32}{3} \pi r^3 - 4\sqrt{3} \pi r^3 = \left( \frac{32}{3} - 4 \sqrt{3} \right) \pi r^3. \]

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