Attempt at Deriving the Volume of a Cone

[mburt]

Hi, so this morning I made an attempt at this... With javascript (website programming language) I was able to successfully yield the ratio of the volume of a cone compared with the volume of a cylinder (1/3).

This is the source code:

<script>
var pi = Math.PI,
cone_v = 0,
h = 0,
r = 1;

for (var i = 0, h = 0;i < r; i = i+0.0001, h++) {
cone_v += pi*i*i;
}

var cyl_v = pi*r*r*h;

var cone_constant = cone_v/cyl_v;

//cone_constant = 1/3
</script>

And basically this is the idea. It's a summation of the area of a circle from radius 0 to 1 (adding on 0.0001 each time), where the height is the number of times the summation is iterated. So in this case it iterates 1000 times... Therefore this is the height, in units.

The volume of the cone = 10473.5 units^2
The volume of the cylinder = 31419.1 units^2

If you divide the volume of cone by cylinder you get approximately one third.

... Now I attempted represent this with calculus but I think I failed. This is as far as I got:

\sum_{0} i = i + \frac{1}{\infty}
therefore
h = 1^\infty

V_cone = \pi(5^\infty)^2 = 25\pi^\infty

V_cylinder = \pi{(1)^2}1^\infty = \pi^\infty

But clearly 25/1 doesn't equal 1/3. What am I doing wrong here? Is it possible to represent this relationship with a summation formula?

**EDIT: accidentally said "area" not volume. Whoops

 

[vector22]

You can solve for the volume of a cone by the disk method.
Let r = h so that r varies as h varies exactly.

Imagine the volume of a cone as stacked disks, the radius of every disk different.

The approx volume of a disk at any r is:

\Delta V = \pi r^2 \, \, \Delta r
The exact volume of a disk at any r is:

dV = \pi r^2 \, \, dr
Integrate to find volume of a cone

V = \int \pi r^2 \, \, dr

=\frac {\pi r^3}{3}

since h = r

=\frac {\pi r^2h}{3}

 

[mburt]

I'm just getting differentiation down now, I guess I need to do some research on integration. Thanks!

And the "disks" method is exactly how I had pictured it, I just didn't know how to describe it