Let B be the solid whose base is the circle x^2+y^2=3^2 and whose vertical cross

sections perpendicular to the x-axis are equilateral triangles.

Compute the volume of B.

Question

Let B be the solid whose base is the circle x^2+y^2=3^2 and whose vertical cross 

sections perpendicular to the x-axis are equilateral triangles. 

Compute the volume of B.

anonymous · Answer

The area of an equilateral triangle of side s is $$ \frac{s^2 \sqrt 3}4 $$ We are going to use this in the proof http://www.mathwords.com/a/area_equilateral_triangle.htm

anonymous · Answer

The cross sections at x will be equilateral triangles of sides 2 y

anonymous · Answer

The  area of each cross section is
$$
\frac{ (2 y)^2 \sqrt 3}4= \sqrt  3\, y^2=\sqrt  3\, \left (9-x^2    \right )
$$

anonymous · Answer

So 
$$
dV=\sqrt  3\, \left (9-x^2    \right ) dx
$$

anonymous · Answer

So $$

V=\int_{-3}^3 \sqrt  3\, \left (9-x^2    \right ) dx=36 \sqrt 3

$$

anonymous · Answer

what is the left and right?

anonymous · Answer

oh wait, i got it, thank you

anonymous · Answer

YW