I'm not sure about this proof of the integral of 2x^2+1 from the vol. 1 of Apostol...

for the case 2x^2+1, the inequality becomes

$\displaystyle \frac{2b^3}{3}-\frac{2b^3}{n}+\frac{bn^2}{3}+bn<A<\frac{2b^3}{3}+ \frac{2b^3}{n}+bn+\frac{bn^2}{3}$

so I'm not sure how to prove this, I can calculate this by the Fundamental Theorem

for the x^2 case you have 3 possibilities that

$\displaystyle A>,<=\frac{b^2}{3}$

but then this changes for the other case