Since $f(x)=x+x^2$ is a polynomial of degree $2$, and Simpson's 1/3 rule integrates any polynomial of degree up to $3$ exactly, the numerical answer should match the exact value of the integral.
Compute the exact integral directly:\[\int_0^1(x+x^2)\,dx=\left[\frac{x^2}{2}+\frac{x^3}{3}\right]_0^1=\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\]
Numerically, $\dfrac{5}{6}=0.8\overline{3}\approx 0.83$, which matches the value obtained from Simpson's rule in the first method, confirming there is no approximation error here because the integrand's degree does not exceed $3$.
\[\boxed{\int_0^1(x+x^2)\,dx\approx 0.83}\]