[Math] Volume Using Integration

Question

"Let $B$ be the solid whose base is the circle $x^2+y^2 =1^2$ and whose vertical cross
sections perpendicular to the $x$-axis are equilateral triangles. Compute the volume of $B$."

Any suggestions? I thought I had set up my integral correctly.

Answer 1

The width of the circle at some $x$ value is given by $$2\sqrt{1-x^2}$$ and so, since the area of the equilateral triangle with side length $l$ is given by $$\frac{\sqrt 3}{4}l^2$$ we have that the volume of the solid is $$V=\int_{-1}^{1}\frac{\sqrt 3}{4}\big(2\sqrt{1-x^2}\big)^2dx$$ or $$V=\int_{-1}^{1}\sqrt 3 (1-x^2) dx$$ Can you evaluate this?