Volume of paraboloid in a cylinder

In summary, the conversation discusses the equation of a parabola that is obtained by taking a cross-section passing through the center of a paraboloid. It also talks about breaking the paraboloid into cylinders and finding the volume of each cylinder. The volume of the paraboloid is then calculated using integration, but it is not half the volume of the cylinder circumscribed by it. The error is in using the equation of the paraboloid to find the volume instead of using the cylinder's volume formula.
  • #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.
 
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  • #2
Hamiltonian299792458 said:
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
 
  • #3
Thanks! o:)
 

Related to Volume of paraboloid in a cylinder

What is a paraboloid?

A paraboloid is a three-dimensional shape that resembles a bowl or a lens. It is created by rotating a parabola around its axis.

What is a cylinder?

A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. It is similar to a can or a tube.

How is the volume of a paraboloid in a cylinder calculated?

The volume of a paraboloid in a cylinder can be calculated by multiplying the area of the base of the paraboloid by the height of the cylinder.

What are the units for measuring volume?

The units for measuring volume are cubic units, such as cubic inches or cubic centimeters.

What are some real-life applications of calculating the volume of a paraboloid in a cylinder?

Calculating the volume of a paraboloid in a cylinder is useful in engineering and architecture, as it can help determine the capacity of a container or the amount of material needed for a structure. It is also used in physics and mathematics for various calculations and equations.

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