Pi (mathematical constant)/Proofs/Student level proof that 22 over 7 exceeds π

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We work out the following integral:

It is possible to divide polynomials in a manner that is analogous to long division of decimal numbers. By doing this it, can be shown that

where −4 is the remainder of the polynomial division.


for n=6, 5, 4, 2, and 0, respectively, one obtains

The following holds

The latter integral is easily evaluated by making the substitution


The integrand (expression under the integral) of the integral in equation (1) 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 is positive. Finally,

which was to be proved.