(adsbygoogle = window.adsbygoogle || []).push({}); A solid has a circular base of radius 2. Each cross section cut by a plane perpendicular to a fixed diameter is an equilateral triangle. Compute the volume of the solid.

First, we find a way to define a the distance of a chord of the circle perpendicular to the fixed diameter. The equation [itex]y=\sqrt(2^2-x)[/itex] from x=-2 to 2 gives half the chord, so 2y is equal to the chord's length. At any point x, the solid's area is an equilateral triangle, so all sides must have length equal to the chord of the circle, or 2y. Now the area of an equilateral triangle with side 2y is equal to [itex](2y)^2\sqrt(3)/4 = y^2\sqrt(3)[/itex]. Substituting for y, we have that [itex]Area(x)=(4-x^2)\sqrt{3}[/itex]. Integrating, we find that

[itex]\int_{-2}^2 A(x) dx=2\int_0^2 \sqrt{3}(4-x^2) dx = \frac{32\sqrt{3}}{3}[/itex]

The problem is that the book has [itex]\frac{16\sqrt{3}}{3}[/itex], and I want to make sure I didn't do it incorrectly.

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# Apostol 2.13 - #15 Cavalieri Solids (Volume Integration)

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