Calculus

The base of a solid is the circle x2 + y2 = 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 sqrt 3

36

18 sqrt 3

18

The answer isn't 18 sqrt 3 for sure.

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  1. Consider the triangle at distance x. Its base is 2y, and its height is y√3. So, adding up the volumes of all those triangles of thickness dx, we have (applying symmetry)

    v = 2∫[0,3] 1/2 * 2y * y√3 dx
    = 2√3 ∫[0,3] y^2 dx
    = 2√3 ∫[0,3] (9-x^2) dx
    = 36√3

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