Split up the integration interval [0, 2] into 10 subintervals:
[0, 1/5], [1/5, 2/5], [2/5, 3/5], ..., [9/5, 2]
The left and right endpoints of the [tex]i[/tex]-th subinterval are, respectively,
[tex]\ell_i=\dfrac{i-1}5[/tex]
[tex]r_i=\dfrac i5[/tex]
with midpoints
[tex]m_i=\dfrac{\ell_i+r_i}2=\dfrac{2i-1}{10}[/tex]
for [tex]1\le i\le10[/tex].
Simpson's rule approximates the definite integral
[tex]\displaystyle\int_0^22\pi e^{-x^2}\sqrt{1+(-2xe^{-x^2})^2}\,\mathrm dx=\int_0^22\pi e^{-x^2}\sqrt{1+4x^2e^{-2x^2}}\,\mathrm dx[/tex]
by interpolating the integrand, [tex]f(x)=2\pi e^{-x^2}\sqrt{1+4x^2e^{-2x^2}}[/tex], over each subinterval with a quadratic polynomial, [tex]p_i(x)[/tex], such that
[tex]\displaystyle\int_0^2f(x)\,\mathrm dx\approx\sum_{i=1}^{10}\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx=\sum_{i=1}^{10}\dfrac{r_i-\ell_i}6(f(\ell_i)+4f(m_i)+f(r_i))[/tex]
You would end up with a value of about 6.497681. Compare to the actual value of the integral, which is closer to 6.497684.
