Solid Shape Volume by Disk Method Problem

[harrietstowe]

Homework Statement

The base of a certain solid is an equilateral triangle of side a, with one vertex at the origin and an altitude along the x-axis. Each plane perpendicular to the x-axis intersects the solid in a square cross section with one side in the base of the solid. Find the volume.

Homework Equations

The Attempt at a Solution

I am pretty unsure about this problem but what I did was:
Let a be one side of this equilateral triangle
Let f(x) = (c*x)/(a*sqrt(3))
Let g(x) = (-c*x)/(a*sqrt(3))
so a*sqrt(3) is the altitude and c is a constant
Solve for c:
tan(60 deg)= (a*sqrt(3))/c
c = a
Sub that into f(x) and g(x)
f(x) = (sqrt(3)*x)/3
g(x) = (-sqrt(3)*x)/3
The next thing I did was to subtract the two functions to get the area
A=f(x)-g(x)= (2*sqrt(3)*x)/3
the lower limit of integration will be 0
the upper limit of integration will be a
So I took the integral (with respect to x) from 0 to a of the Area squared and I got (4a^3)/9
The correct answer though is (sqrt(3)*a^3)/6
I hope my set up wasn't completely off but I guess you guys can give me so feedback.
Thanks

 

[ehild]

The integration limit is not correct. If you made a drawing and saw the picture of the body you would have found out the correct limits of integration.

ehild