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:
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]