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 $\dfrac{\sqrt3}4s^2$.

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

$x^2+y^2=9\implies y=\pm\sqrt{9-x^2}$

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

$\sqrt{9-x^2}-\left(-\sqrt{9-x^2}\right)=2\sqrt{9-x^2}$

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

$\dfrac{\sqrt3}4(2\sqrt{9-x^2})^2=\sqrt3(9-x^2)$

and the volume of the solid is

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