The base of a solid in the xy-plane is the circle x^2 + y^2 = 16. Cross sections of the solid perpendicular to the y-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

Question

answer 1: (4√3)/3
answer 2: (64√3)/3
answer 3: (256√3)/3
answer 4: 256pi√3

Answer 1

in quadrant where x and y are positive
base = x
height = x sqrt3
area = (1/2) x^2 sqrt 3
integrate area dy from y = 0 to y = 4 and multiply by 4
2 sqrt3 integral (16-y^2)dy
2 sqrt 3 [16 y - y^3/3]
2 sqrt 3 (16)(4 - 4/3)
32 sqrt 3 * 8/3
(256 /3 ) sqrt 3