The graph of … has a sharp spike and a long tail

The graph of $y(x)=1/(x^2+10^{-10})$ has a sharp spike and a long tail. Estimate ${\int }_0^{1}ydx$ from $T_{10}$ and $T_{100}$ (don't expect much). Then substitute $x=10^{-5}\tan \theta$, $dx=10^{-5}{\sec }^2\theta d\theta$ and integrate $10^5$ from 0 to $\pi /4$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.8 Numerical Integration, #23