So, writing things as we have done so,

with the apex of the origin of the plane.

We see that the equation for

the line that determines the profile of the cone is y = h/r x.

Solving for x as r over h times y, we substitute into the formula for

the volume element to obtain pi times quantity

(r/h y) squared dy.

And now to compute the volume, we integrate this

volume element as y goes from 0, the bottom to h, the top.

Pulling out the associated constants gives pi r squared/h squared.

What's left is the integral of y squared dy, which is, of course, y cubed over 3.

Evaluating from 0 to h yields the perhaps

familiar formula 1/3 pi r squared h.

Now what happens if,

instead of a cone with a circular base, we consider something else.

Let's say a square, side length s.

Proceeding as before by turning the cone upside down and

slicing along lateral planes.

We see that the volume element is a thickened square of some side length.

What is that side length?

If we solve for the line that constrains y in terms of x and solve for

x, then we obtain a volume element that is 4 x squared dy.

Namely, 4 times quantity (s/2h y) squared dy.

Integrating that to obtain the volume leads to the integral

of s squared over h squared times y squared dy.

As y goes from zero to h.

Pulling out the constants we are left with the integral of y squared dy,

yielding y cubed/3.

Evaluating from zero to h and multiplying by those constants out in front

leads to 1/3 s squared h.

You may sense a pattern here.

In both cases, it was one-third the area of the base times the height.

Well, it turns out that this is true in general for

any cone with a base of area B, and height h.

No matter what its shape, we can, by turning it upside down and

slicing, obtain copies of that base.

Copies that are rescaled by some factor.

So that the volume element is B, the area of the base,

times some factor, times the thickness dy.

The question is, what is that scaling factor as a function of y?

Well, let's say that you go halfway down.