Volume of Cone

series 1: Volume of Cone

We will calculate volumes by cross sections with helps of definite integral.

Let’s first look at the type of cross section in horizontal:

Let’s denote radius of cross section of “y”, now we are going to figure out formula of it:

The above procedure seems nice & neat.

Now let’s challenge ourselves by asking:

Why should us calculate the volume by figuring out the “y”, instead of “x”?

Or, why shouldn’t calculate by cross sections of vertical?

Compare with previous one:

So, where is the error?

Following is the cite of Conic Section from wikipedia:

So do not mislead by the previous wrong infer, the vertical section is hyperbola instead of triangle.

Or think in this way:

We are actually calculating the volume of a pyramid, just thinking the radius of “r” is a special case of a pyramid:

We may get the area of pyramid the above case is: “ r * r * 2 “. So the volume turns from:

into:

in which S is the area of bottom of pyramid, it’s also the formula of volume of pyramid.

Math is fun.~~~~

[post status: almost done]