3.19: Shaded area; log AP and GP conditions

The diagram shows part of the curve $y=3+\dfrac{4}{2x+1}$ and the straight line $3y=2x+6$. Find the area of the shaded region, giving your answer in exact form.

An AP has first three terms $(2x+1)$, $4(2x+1)$, $7(2x+1)$ with $2x+1\neq0$.

Show $S_{n}=\dfrac{n}{2}(2x+1)(3n-1)$.

Given that the sum to $n$ terms is $(54n+37)(2x+1)$, find the value of $n$.

Given also that the sum to $n$ terms in part (ii) is equal to $1017.5$, find the value of $x$.

A GP has first term $(2y+1)$ and common ratio $3(2y+1)$. Given that $u_{n}=4u_{n+2}$, find the possible values of $y$.

A GP has first term $\mathrm{sin}\,\theta$ and common ratio $2\mathrm{sin}^{2}\theta$, where $0<\theta<\dfrac{\pi}{2}$. Find the set of values of $\theta$ for which the sum to infinity exists.