Hello, coe236!
$\displaystyle A$ is the area bounded by: .$\displaystyle y\:=\:x^2\text{ and }y\:=\:1$
Suppose a solid has a base A and the crosssections of the solid
perpendicular to the yaxis are equilateral triangles.
Sketch solid and find volume.
The base looks like this: Code:

1
*+*

x  x
*    +    *
*  *
*  *
     *      

The area of an equilateral triangle with side $\displaystyle s$ is: .$\displaystyle A \:=\:\frac{\sqrt{3}}{4}s^2$
The side of our equilateral triangle is: .$\displaystyle 2x$
Hence, the area of the triangle is: .$\displaystyle A \:=\:\frac{\sqrt{3}}{4}(2x)^2 \:=\:\sqrt{3}\,x^2$
Since $\displaystyle y \,=\,x^2$, we have: .$\displaystyle A \:=\:\sqrt{3}\,y$
Therefore, the volume is: .$\displaystyle V \;=\;\sqrt{3}\int^1_0 y\,dy$