Pi (mathematical constant)/Proofs/Student level proof that 22 over 7 exceeds π: Difference between revisions
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We work out the following integral: | We work out the following integral: | ||
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Revision as of 05:26, 16 September 2009
We work out the following integral:
One can divide polynomials in a manner that is analogous to long division of decimal numbers. By polynomial division one shows that
where −4 is the remainder of the polynomial division.
One uses:
for n=6, 5, 4, 2, and 0 and one obtains
The following holds
The latter integral is easily evaluated by making the substitution
The integrand (expression under the integral) of the integral I is everywhere positive on the integration interval [0, 1] and, remembering that an integral can be defined as a sum of integrand values, it follows that the integral I is positive. Finally,
which was to be proved.