Volume of paraboloid in a cylinder

[Hamiltonian]

TL;DR Summary

I want to prove that the volume of a paraboloid is half the volume of the cylinder circumscribed by it.

the equation of a parabola that is obtained by taking a cross-section passing through the center of the paraboloid is ##y = ax^2##

breaking the paraboloid into cylinders of height ##(dy)## the volume of each tiny cylinder is given by ##\pi x^2 dy##
since ##y = ax^2## we have ##\pi (y/a) dy##

now on integrating this $$V = \int_0^h \pi (y/a) dy = \frac{\pi h^2}{2a} + c$$

the answer I have got for the volume of the paraboloid is not half the volume of the cylinder circumscribed by it.
I have a feeling I am doing something majorly wrong, as I think you are supposed to use the equation of a paraboloid to find the volume but I am not too sure about that.

 

[LCSphysicist]

Summary:: I want to prove that the volume of a paraboloid is half the volume of the cylinder circumscribed by it.

the equation of a parabola that is obtained by taking a cross-section passing through the center of the paraboloid is ##y = ax^2##

breaking the paraboloid into cylinders of height ##(dy)## the volume of each tiny cylinder is given by ##\pi x^2 dy##
since ##y = ax^2## we have ##\pi (y/a) dy##

now on integrating this $$V = \int_0^h \pi (y/a) dy = \frac{\pi h^2}{2a} + c$$

the answer I have got for the volume of the paraboloid is not half the volume of the cylinder circumscribed by it.
I have a feeling I am doing something majorly wrong, as I think you are supposed to use the equation of a paraboloid to find the volume but I am not too sure about that.

The cylinder's volume is $$\pi h r^2$$, you are calling r as x, $$V = \pi h x^2$$, $$V = \frac{\pi h y}{a}$$, but, obviously y = h, $$V = \frac{\pi h^2}{a}$$. You error is thinking you are wrong

 

[Hamiltonian]

Thanks!