Question

A cylinder has radius HC and height h.
What is the ratio of the volume of the paraboloid to the volume of the cylinder?

Answer 1

this the diagram

Answer 2

the paraboloid of y=x^2 have rounded x-axes or y-axes ?

Answer 3

y-axis and the volume for the parabloid is pih^2/2

Answer 4

yeps... the volume of parabolid :
Vp = pi * int (sqrt(y))^2 dy [0,h]
= pi * int (y) dy [0,h]
= pi * 1/2* y^2 [0,h]
= pi*1/2*h^2
Next, to find the volume of cylinder, first we must determine the intersection between line y=h and y=x^2
x^2=h
x=+-sqrt(h) (nevermind the negative sign, because radius impossible -)
so, the radius of cylinder r=sqrt(h)
just doing by manual, :p
Vc = pi*r^2*t
= pi*(sqrt(h))^2*h
=pi*h^2
now u can get ratio of both :
Vp : Vc = ....