f(x) = 2/(1+x**2) I = \int_{-1}^{1} f(x) dx = pi Trapezoidal rule: N # x_k I_N I_N - I 1 2 2.000000000000 -1.141592653590 2 3 3.000000000000 -0.141592653590 4 5 3.100000000000 -0.041592653590 8 9 3.131176470588 -0.010416183002 16 17 3.138988494491 -0.002604159099 32 33 3.140941612041 -0.000651041548 Simpson's rule: N # x_k I_N I_N - I 1 2 2.000000000000 -1.141592653590 2 3 3.333333333333 0.191740679744 4 5 3.133333333333 -0.008259320256 8 9 3.141568627451 -0.000024026139 16 17 3.141592502459 -0.000000151131 32 33 3.141592651225 -0.000000002365 Gauss Legendre quadrature: # x_k I_N I_N - I 1 4.000000000000 0.858407346410 2 3.000000000000 -0.141592653590 3 3.166666666667 0.025074013077 4 3.137254901961 -0.004337751629 5 3.142342342342 0.000749688753 6 3.141463414634 -0.000129238956 7 3.141614906832 0.000022253243 8 3.141588825092 -0.000003828498 9 3.141593311880 0.000000658290 10 3.141592540447 -0.000000113143 11 3.141592673030 0.000000019441 12 3.141592650250 -0.000000003340 13 3.141592654163 0.000000000574 14 3.141592653491 -0.000000000098 15 3.141592653607 0.000000000017 16 3.141592653587 -0.000000000003 17 3.141592653590 0.000000000000 18 3.141592653590 0.000000000000