base of S is the parabolic region {(x,y) x^2<y<1}.Cross sections perpendicular to y-axis are equilateral tri

Question

What is the volume

Answer 1

At a distance x from the point (1,1) of the parabolic region the side of the triangle is 1-x^2. The height is 1/2sqrt(3)*(1-x^2) and the
surface is S = 1/4sqrt(3) (1-x^2)^2
As the problem is symmetric the volume is
V =2*1/4*sqrt(3)Int (0,1) (1-x^2)^2dx=1/2sqrt(3)(1-2/3+1/5)=
4/15*sqrt(3)

Answer 2

V=2
https://www.youtube.com/watch?v=BtaNbEtv6-Q
Here s the link of the steps