Mathematics - Definite Integration Question with Solution | TestHub

For decreasing $q(t)=1/\left(1+t^2\right)$ on $[0,1/\sqrt{3}],X$ and $Y$ are respectively the right and left Riemann sums. Thus $X<{\int }_0^{1/\sqrt{3}}q(t)dt=\pi /6<Y$. Also $X+Y<\pi /3$by pairing the endpoint errors, as in the trapezoidal bound for this concave portion.