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Apostol 2.13  #15 Cavalieri Solids (Volume Integration) 
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#1
Aug1911, 06:27 PM

P: 106

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(2^2x)[/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)=(4x^2)\sqrt{3}[/itex]. Integrating, we find that [itex]\int_{2}^2 A(x) dx=2\int_0^2 \sqrt{3}(4x^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. 


#2
Aug1911, 06:42 PM

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P: 1,664

Check the area of your triangular crosssections again. If the base is 2y , what is the height?



#3
Aug1911, 07:00 PM

P: 106

This agrees with the formula for the area of an equilateral triangle given here: http://www.mathwords.com/a/area_equi...l_triangle.htm Taking [itex]s=2y[/itex], we have that the area is equal to [itex]\frac{(2y)^2\sqrt{3}}{4}=y^2\sqrt{3}[/itex]. 


#4
Aug1911, 07:13 PM

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P: 1,664

Apostol 2.13  #15 Cavalieri Solids (Volume Integration)
Sorry, yes: my fault for trying to deal with more than one matter at once. I am wondering if the solver for Apostol used symmetry and forgot to double the volume integration. I am getting the same answer you are.
Stewart does this as Example 7 in Section 6.2 with a radius of 1 and gets oneeighth our volume, which is consistent. Backofthebook answers aren't 100%... 


#5
Aug1911, 07:20 PM

P: 106




#6
Aug1911, 07:25 PM

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What edition is Apostol up to now? Generally, Third and later Editions have the error rates in the answer sections down to about 0.25% or less...



#7
Aug1911, 07:28 PM

P: 106




#8
Aug1911, 07:48 PM

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P: 1,664

Well, it's supposed to be a classic. But I suspect the percentage of errors in the answers could be somewhere in the 0.25% to 0.5% range (from my long experience with textbooks)...
I looked Apostol up and he's 88 this year. I doubt he's going to revise the book (though I've been surprised in the past); he's moved on to other projects. 


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