Join TSR
 
 About Us | FAQs | Sign in
 
HomeForumsArticlesRevision NotesPersonal StatementsUniversity GuidesHealth & RelationshipsPostgraduate StudyCareers
Join the UK's largest student community  
Advanced
Search

Join The Student Room Today

Be part of the UK's largest and fastest growing student community.

It's free to join and a lot of fun - Get inspired, express your ideas, interact and share

Join The Student Room Today

STEP I 1997 question 6 solution

From The Student Room

TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 1997 question 6 solution

\displaystyle x^4(1-x)^4 = x^8-4x^7+6x^6-4x^5+x^4. The easiest way of doing the next bit would probably be long division, but that will be painful to layout in TeX. So, let's compare coefficients...


Coeff of \displaystyle x^8: a_6 = 1

Coeff of \displaystyle x^7: a_5 = -4

Coeff of \displaystyle x^6: a_6+a_4 = 6 \implies a_4 = 5

Coeff of \displaystyle x^5: a_5+a_3 = -4 \implies a_3 = 0

Coeff of \displaystyle x^4: a_4+a_2 = 1 \implies a_2 = -4

Coeff of \displaystyle x^3: a_3+a_1 = 0 \implies a_1 = 0

Coeff of \displaystyle x^2: a_2+a_0 = 0 \implies a_0 = 4

Coeff of \displaystyle x^1: a_1 = 0 (Which we knew it should be, but it never hurts to check).

Coeff of \displaystyle x^0: a_0+b = 0 \implies b=-4


Using this in the integral we get:

\displaystyle \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx = \int_0^1 x^6-4x^5+5x^4-4x^2+4 - \frac{4}{1+x^2} dx

RHS = \displaystyle \frac 17 - \frac 46 + 1 - \frac 43 + 4 - \pi = \frac {22}{7} - \pi as required.

[Unparseable or potentially dangerous latex formula. Error 5 : 951x49]

But \displaystyle 1+x^2 > 1 \text{ for } x > 0 and so \displaystyle 0 < \int_0^1 \frac{x^4(1-x)^4}{1+x^2} < \int_0^1 x^4(1-x)^4 = 1 / 630.

So since we know \displaystyle \pi = \frac{22}{7} - \frac{x^4(1-x)^4}{1+x^2} we deduce \displaystyle \frac {22}{7} > \pi > \frac{22}{7} - \frac{1}{630}.

Solution by DFranklin

Retrieved from "http://thestudentroom.co.uk/wiki/STEP_I_1997_question_6_solution"
Article Tools:  Create New Article  |  Report This Article  |  This Article's History  |