This is a pair of similar questions.
Here is the first one:
Does there exist a rational function $P(x)$ with a,b ∈ ℚ such that $\int_a^b P(x)\operatorname{d}x$ is equal to $k\cdot e$ for some $k\in\Bbb{Q}$ 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 $x^2+y^2 < r$ for all $(x,y)\in A$ for some finite $r$. Does such an $A$ exist such that the area of $A$ equals $k\cdot e$ where $k\in\Bbb{Q}$ 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 $x^2+y^2 < 1$. 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.
I have now restated this question more clearly Creating e using rational functions. Thanks for the help fixing up the wording.
I'm assuming $a$ and $b$ are fixed rational numbers.
If $P(x) = N(x)/D(x)$ is a rational function with rational coefficients for numerator and denominator, then the roots of the denominator $D(x)$ are algebraic numbers. We can perform a partial fraction decomposition of $P(x)$ and integrate it from $x=a$ to $x=b$, obtaining a result that is a linear combination of $1$ and natural logs of algebraic numbers with algebraic coefficients. The question is whether $e$ can be expressed in this way.
Unfortunately this is still an open question, though I believe it would follow from Schanuel's conjecture that it can't be so expressed. We don't even know that $e + \pi$ is irrational. If $e + \pi = r$ was rational, you could write $$e = \int_0^1 \left(r - \frac{4}{1+x^2}\right)\; dx$$