consider A, if A is the area bounded by y=x^2 and y=1. Suppose a solid has a base A nd the cros sections of the solid perpendicular to the yaxis are equilateral triangles. sketch solid nd find volume.

I know how to go about finding the area of A by evaluating the integral and I understand the concept of finding vol. by using cross sections but I dont know how to approach or start this problem. Can someone help please?

2. Hello, coe236!

$A$ is the area bounded by: . $y\:=\:x^2\text{ and }y\:=\:1$
Suppose a solid has a base A and the cross-sections of the solid
perpendicular to the y-axis 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 $s$ is: . $A \:=\:\frac{\sqrt{3}}{4}s^2$

The side of our equilateral triangle is: . $2x$

Hence, the area of the triangle is: . $A \:=\:\frac{\sqrt{3}}{4}(2x)^2 \:=\:\sqrt{3}\,x^2$

Since $y \,=\,x^2$, we have: . $A \:=\:\sqrt{3}\,y$

Therefore, the volume is: . $V \;=\;\sqrt{3}\int^1_0 y\,dy$