• Jan 24th 2008, 07:41 AM
coe236
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?
• Jan 24th 2008, 09:45 AM
Soroban
Hello, coe236!

Quote:

$\displaystyle A$ is the area bounded by: .$\displaystyle y\:=\:x^2\text{ and }y\:=\:1$
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 $\displaystyle s$ 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: .$\displaystyle V \;=\;\sqrt{3}\int^1_0 y\,dy$