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       From Euclid to Spigotry
Assignment 13 (outline 12)
Last modified: September 14, 2004

1. Newton could have chosen to consider the area of one-fourth of a circle or one-eighth of a circle instead of one-sixth of a circle.  Both these alternatives are illustrated in the diagram to the right.
For each alternative, find a formula for p that involves , specifying the upper limit of integration a, and determine what estimate for p you get by keeping just the first two terms of the series from topic 4 of the notes.

2. Instead of using the arc tangent function to compute p, we could use the arcsin function.
The relevant integral is arcsin x = 
a. Get the first five terms in the binomial expansion of  (Beckmann has 4 terms on p.141)
b. Get the first five terms in the series for arcsin x.
c. Using Mathematica, sum the first five terms of the series for arcsin  . Multiply the answer by 6,
and you should get something close to p.