Question

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

Answer 1

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}`