This is a pair of similar questions. Here is the first one: Does there exist a rational funct

fecundavai3c

fecundavai3c

Answered question

2022-02-15

This is a pair of similar questions.
Here is the first one:Does there exist a rational function P(x) with a,b ϵQ such that abP(x)dx is equal to ke for some kϵQ where e is Euler's number? Prove or disprove that such a function exists. If such a function exists find it. Also, this function should be bounded in the y dimension also so there should exist some r such that |P(x)| < r for every x on [a,b].
This is the second:Let A be a region defined by the inequality 0<P(x,y) where P is a rational function and x2+y2<r for all (x,y)ϵA for some finite r. Does such an A exist such that the area of A equals ke where kϵQ and e is Euler's constant? Prove or disprove that such an A exists. If such an A exist find P.
Inspiration for the questions:A nice way to approximate pi is by picking random points in the region 0<x<1 and 0<y<1 and seeing if x2+y2<1. This is easy to program because all the functions involved are rational. similar things can be done with ln(2) for example. I would like to do so with e but can't think of the correct shape to use.
The coefficients of the rational functions should be rational numbers. The bounds on a and b in the first question should be rational.

Answer & Explanation

ofisu2n3

ofisu2n3

Beginner2022-02-16Added 4 answers

Im
sergiotheguyqgs

sergiotheguyqgs

Beginner2022-02-17Added 4 answers

Thank you!

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