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Compute the volume of a solid with given properties

Given a solid with circle base of radius 2 and cross sections which are equilateral triangles, compute the volume of the solid.

We may describe the top half of the circular base of the solid by the equation

    \[ f(x) = \sqrt{4-x^2}. \]

Thus, the length of the base of any equilateral triangular cross section is

    \[ 2 \sqrt{4-x^2} \qquad -2 \leq x \leq 2. \]

Since these are equilateral triangles with side length 2 \sqrt{4-x^2}, the area is given by

    \[ A(x) = \sqrt{3}(4-x^2). \]

Then we compute the volume,

    \begin{align*}  V &= \int_{-2}^2 \sqrt{3}(4-x^2) \, dx \\    &= 4\sqrt{3}(4) - \sqrt{3} \left( \left. \frac{x^3}{3} \right|_{-2}^2 \right) \\    &= 16 \sqrt{3} - \frac{16}{3} \sqrt{3} \\    &= \frac{32 \sqrt{3}}{3}. \end{align*}

(Note: Apostol gives the solution \frac{16 \sqrt{3}}{3} in the back of the book, but I keep getting \frac{32 \sqrt{3}}{3}, as does Edwin in the comments. I’m marking this as an error in the book for now. If you see where my solution is wrong and Apostol is correct please leave a comment and let me know.)