Volume of a solid with equilateral triangle cross-sections

[Phyzwizz]

The base of a solid is the region bounded by the parabola x² = 8y and the line y = 4 and each plane section perpendicular to the y-axis is an equilateral triangle. What is the volume of the solid?(Barron's Problem)

so I solved for x since y must be used because the cross section is perpendicular to the y-axis. x=(8y)^1/2. The area equation for an equilateral triangle is A = s²√(3)/4
So I figured the volume of the cross section would be equal to the integral from 0 to 4 of (8y√(3))/4 which gives me 16√(3) which is apparently wrong. The answer is supposed to be 64√(3). I don't know what I did wrong.

 

[HallsofIvy]

8²= 64, not 8!

 

[Phyzwizz]

8²= 64, not 8!

I wrote down solving for x incorrectly its supposed to be √(8y) which allows my confusion to continue.