- #1

Hamiltonian

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- 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.

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.