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

The metadata subpage is missing. You can start it via filling in this form or by following the instructions that come up after clicking on the [show] link to the right.

Do you see this on PREVIEW? then SAVE! before following a link. To find out more about the use of metadata for the management of Citizendium pages, see CZ:Metadata. You may see this box for one of two reasons. Either: A - New page has been created, or subpages template was added to an existing page B - Cluster move is in progress Click either the A or B link for further instructions. [Feel free to recommend improvements to the text or links in this template] A - For a New Cluster use the following directions Subpages format requires a metadata page. Using the following instructions will complete the process of creating this article's subpages. Click the blue "metadata template" link below to create the page. On the edit page that appears paste in the article's title across from "pagename =". You might also fill out the checklist part of the form. Ignore the rest. For background, see Using the Subpages template Don't worry--you'll get the hang of it right away. Remember to hit Save! the "metadata template". However, you can create articles without subpages. Just delete the {{subpages}} template from the top of this page and this prompt will disappear. :) Don't feel obligated to use subpages, it's more important that you write sentences, which you can always do without writing fancy code. [Feel free to recommend improvements to the text or links in this template] B - For a Cluster Move use the following directions The metadata template should be moved to the new name as the first step. Please revert this move and start by using the Move Cluster link at the top left of the talk page. The name prior to this move can be found at the following link. [Feel free to recommend improvements to the text or links in this template]

---

We work out the following integral:

$I\equiv \int _{0}^{1}{\frac {t^{4}(1-t)^{4}}{1+t^{2}}}\,\mathrm {d} t$

One can divide polynomials in a manner that is analogous to long division of decimal numbers. By polynomial division one shows that

${\frac {t^{4}(1-t)^{4}}{1+t^{2}}}={\frac {t^{8}-4t^{7}+6t^{6}-4t^{5}+t^{4}}{1+t^{2}}}=t^{6}-4t^{5}+5t^{4}-4t^{2}+4-{\frac {4}{1+t^{2}}}$

where −4 is the remainder of the polynomial division.

One uses:

$\int _{0}^{1}t^{n}\,\mathrm {d} t={\frac {1}{n+1}}$

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

$\int _{0}^{1}(t^{6}-4t^{5}+5t^{4}-4t^{2}+4)\,\mathrm {d} t={\frac {1}{7}}-{\frac {4}{6}}+{\frac {5}{5}}-{\frac {4}{3}}+{\frac {4}{1}}={\frac {22}{7}}$

The following holds

$-4\int _{0}^{1}{\frac {1}{1+t^{2}}}\,\mathrm {d} t=-4\left[\arctan(t)\right]_{0}^{1}=-4{\frac {\pi }{4}}=-\pi$

The latter integral is easily evaluated by making the substitution

$t=\tan(x)\,$

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

$0<I={\frac {22}{7}}-\pi \quad \Longrightarrow \quad {\frac {22}{7}}>\pi$

which was to be proved.