-
-
--------------------- 2516 - CALCULUS - Calculus of a Circle?
-
- How did the ancients define the area of a circle to be pi*r^2? Well, first of all they had no TV’s, so they did not have to watch American Idol, or Dr. Phil, or Entertainment Tonight. They had no electricity so they sat by candlelight and had time to think.
-
- They figured out that all circles , regardless of their size, had the same ratio of circumference to diameter. Let’s call it c / d.
-
- And the circumference of any circle must be circumference = d*c / d = c.
-
--------------------- Or, c = 2*r*c / d.
-
- Let’s take a circle and divide it into wedges just like you would cut up a pizza pie. Now take each wedge and stack them side by side with a point up, then a point down, then up, then down, etc. until you have formed a parallelogram.
-
- If you cut the pie in small enough wedges and stack them this way it will look more and more like a rectangle. And the area of a rectangle is length times height. Cut the pie into a million pieces and you have a rectangle for sure. Area = length * height.
-
------------------------ But, the height is radius, r.
-
------------------------ And, the length is half the circumference, ½*c.
-
------------------------ The circumference is 2*r*c/d.
-
------------------------ So, the Area = r*½* 2*r*c/d
-
------------------------ Or, Area = c/d * r^2
-
- OK, now all we have to do is define c / d as pi, after our pizza pie.
And, Area = pi*r^2
-
- Summing up all these pieces became known as integration and Calculus. And, you can do amazing things with the mathematics of Calculus. Pi became calculated as 3.14159.….. the calculation goes to infinity.
-
- If you take the derivative of the area of a circle you get the circumference,
d(pi*r^2) /dr = 2*pi*r ( in the math tables it is d(x^n) = n*x^(n-1)* dx).
-
- So, like the velocity is the rate of change of distance with respect to time. d(v) = dx/dt The area is the rate of change of the circumference with respect to the radius. d(A) = dc/dr
The integral of the circumference with respect to the radius is the area.
-
- The intergal of 2*pi*r = 2*pi*r^2 / 2 = pi*r^2. ( in the math table the integral of x^n*dx = x^(n+1) /( n+1)
-
- This makes sense, it you start with a circle with 1 inch radius you get a circumference of 6.28 inches. And it is a hollow circle. But, if you start with a point and you integrate, or do a summation, of each circumference out to 1 inch radius you get a solid circle with an area of 3.14 square inches.
-
- This same process of derivatives and integrals works for the volume of a sphere, 4/3*pi*r^3, to the surface area of a sphere, 4*pi^r^2.
-
- TO SUMMARIZE:
------------------ The perimeter of a circle is the rate of change or area with radius. Rate of change is the slope, or the derivative of area with radius. The derivative of pi*r^2 = 2*pi*r. The general formula for the derivative of x^n * dx = x^n-1
-
----------------- The area of a circle is the summation of the perimeters with radius. Summation is integration of perimeter with radius. The integration of 2*pi*r = p*r^2.
The general formula for the integration of x*dx = x^n+1 / n+1
-
------------------ The surface area of a sphere is the rate of change or volume with radius. Rate of change is the slope, or the derivative of volume with radius. The derivative of 4/3*pi*r^3 = 4*pi*r^2. The general formula for the derivative of x^n * dx = x^n-1
-
----------------- The volume of a sphere is the summation of the surface areas with radius. Summation is integration of surface areas with radius. The integration of 4*pi*r^2 = 4/3*pi*r^3. The general formula for the integration of x*dx = x^n+1 / n+1
-
- Pi is an irrational number, which means it can not be written as a ratio of two integers. 22/7 is only accurate to 2 decimal points. 355/113 is accurate only to 6 decimal places.
-
- In 1657 pi originally stood for “peripheral / diameter” ( I was just kidding about the pizza pie). In 1600 Ludolph van Ceulen calculated pi out to 35 decimals. He was so proud of what he had done he had it engraved on his tombstone. The symbol for pi was first used in 1706.
-
- In 1789 pi was extended out to 140 places ( of which it was later found that only 137 places were correct). In 1873 it was extended to 707 places and that stood until 1949 when the computer was invented. Today it is out to 1,240,000,000 places.
-
- 3.1415926535897932384626433832795028841971.……… to infinity
-
- You can remember some of these decimal places by remembering “ How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics” = 3.14159265358979. Count the letters and you have pi to 14 decimal points.
-
- Pi shows up in a lot of places. See Review 513 “Buffon’s Needles”, which tell how repeatedly dropping breadsticks on a tile floor can be used to statistically calculate the value of pi.
-
- See Review 400 “Life is Uncertain“, Heisenberg’s uncertainty principle for delta position * delta momentum = Planck’s constant / 4*pi.
-
--------------------- Or, Einstein’s field equation: R-1/2gR + delta g = 8*pi*G^4*T.
-
---------------------- The series 1/1^2 + ½^2 + 1/3^2 + … = pi^2/6
-
---------------------- The series 1/1-1/3+1/5-1/7+1/9-…… = pi/4
-
---------------------- e^i*pi +1 = 0
-
- The great pyramids in Egypt were constructed with the ratio of the perimeter of the base to twice the height = pi.
-
- How all these things have to do with circles I can not imagine. But the ratio of the circumference to the diameter sure shows up in a lot of places. It all must be connected somehow???
-
- Other Reviews available upon request:
-
- 513 “Buffon’s Needles”, which tell how repeatedly dropping breadsticks on a tile floor can be used to statistically calculate the value of pi.
-
- 400 “Life is Uncertain“, Heisenberg’s uncertainty principle for delta position * delta momentum = Planck’s constant / 4*pi
-
- 2514 - Everyday Calculus
-
- 734 - Spacetime, Energy, and Calculus
-
- December 1, 2019. 2516 745
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Sunday, December 1, 2019 -------------------------
-----------------------------------------------------------------------------------------
No comments:
Post a Comment