Question

I really don't understand this calculus 2 problem?

The question is: An equilateral hexagon is revolving around one of its edges. Find the volume of the solid of revolution

Answer 1

`V = pi l^3`

Explanation:

Considering `l` as the equilateral hexagon side

`h=l cosphi`
`delta=lsinphi`

we have

`V =2 1/3 pi h^2 delta + pi h^2 l = (2/3cos^2phisin phi+cos^2phi)pi l^3`

but `phi=pi/3` so `sin phi = 1/2` and `cos phi = sqrt3/2` and finally

`V = pi l^3`

Answer 2

Pappus 2nd Theorem: Volume V of a solid of revolution generated by rotating a plane figure about an external axis is equal to the area of the figure times the distance traveled by its geometric centroid, or `V = A d`

`A` is calculated in the figure as `6 xx 1/2 b h`

`d = 2 pi (s sqrt3/2)`

`implies V = ( 9pi)/2 s^3`

Answer 3

I have solved this way, using integrals, as shown below: