How does the magic work?

Question

I have a magical formula for volume of any solid :
V = 1/6 (bottom base + 4 midsection + top base) * height. 
It works like a Swiss clockwork.

Volume of pyramid:
bottom base area = S
midsection area = S/4 
top base area (the tip) =0
V = 1/6 (S + 4 S/4 + 0) h = 1/3 Sh
correct

Volume of a brick:
correct

Volume of a tetrahedron standing on an edge
http://mathcentral.uregina.ca/QQ/database/QQ.09.98/jane1.1.gif
bottom base area (the bottom edge) = 0
midsection area = 1/2 
top base area (the top edge) 0
V = 1/6 (0 + 4 * 1/2 + 0) = 1/3 
correct


Volume of a sphere
bottom base area (the South pole) = 0
midsection area = π
top base area (the North pole) 0
V = 1/6 (0 + 4 * π + 0) * 2 = 4/3 π 
correct

Volume of a hemi-sphere
bottom base area (equatorial) = π
midsection area = 3/4 π
top base area (the North pole) 0
V = 1/6 (0 + 4 * 3/4 π + π) = 2/3 π 
correct

Where is the magic?

Anonymous · Accepted Answer

The volume of a solid can be computed by

V = integral [A(s)] ds,

where A is the area of the intersection of the solid with the plane z = s (that is, the cross-sectional area at a particular height s), and the integral is taken over the interval on which A(s) is nonzero.

The formula that you have provided is a Simpson's Rule approximation to this integral, using two subintervals.

Simpson's Rule with two subintervals gives the exact value of the integral if the integrand is a cubic function (or a quadratic, linear, or constant function).  This is not hard to prove--just evaluate the integral from p to q of ax^3 + bx^2 + cx + d directly and using Simpson's Rule with two subintervals, and you'll see that you get the same result each way.

So your formula gives the exact volume for any solid where its cross-sectional area is an at-most-cubic polynomial function of the position at which we are taking the cross-section.

This means that your formula works for most common solids (sphere, hemisphere, cone, frustum, cylinder, parallelepiped, pyramid, any prism, etc.)

---

A question about this formula was previously asked (a while ago) by Scythian1950, and received several answers which may offer more insight than mine: http://answers.yahoo.com/question/index;_ylt=AoxrW.rXXLlAh4A5tS2bC70jzKIX;_ylv=3?qid=20071010210151AAE41mP

Anonymous · Answer

Elmer Fudd: "i will do it with my spear and magic helmet!" insects Bunny: "Your spear and magic helmet?" Fudd: "Spear and magic helmet!" insects: "Magic helmet?" Fudd: "Magic helmet!" insects (skeptical): "Magic helmet." Fudd singing: "confident, magic helmet! and that i will supply you a pattern!"

Anonymous · Answer

By people WANTING to Believe In it.

Hemant · Answer

Smirnoff, you took my breath away !!!

Anonymous · Answer

If it doesnt work then it doesnt work.

If it does work then someone else already came up with it.

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How does the magic work?

5 Answers