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The base of a solid is the circle x^2 + y^2 = 9. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

36 times the square root of 3
36
18 times the square root of 3
18

User Ssokolow
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1 Answer

4 votes

Recall that the area of an equilateral triangle with side length
s is
\frac{\sqrt3}4s^2.

In the
x-y plane, the base is given by two equations:


x^2+y^2=9\implies y=\pm√(9-x^2)

so that for any given
x, the vertical distance between the two sides of the circle is


√(9-x^2)-\left(-√(9-x^2)\right)=2√(9-x^2)

and this is the side of length of each triangular cross-section for each
x. Then the area of each cross-section is


\frac{\sqrt3}4(2√(9-x^2))^2=\sqrt3(9-x^2)

and the volume of the solid is


\displaystyle\int_(-3)^3\sqrt3(9-x^2)\,\mathrm dx=\boxed{36\sqrt3}

User YYY
by
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