Simpson의 공식(n = 2):

S_{2} = \dfrac{\Delta x}{3}\,[\,f(x_{0}) + 4f(x_{1}) + f(x_{2})\,]

1) 구간 분할

\Delta x = \dfrac{1-0}{2} = 0.5,\quad x_{0}=0,\; x_{1}=0.5,\; x_{2}=1

2) 함숫값

f(x)=\sin\!\left(\dfrac{\pi}{2}x\right)

f(0)=0,\; f(0.5)=\dfrac{\sqrt{2}}{2},\; f(1)=1

3) Simpson 공식 대입

S_{2} = \dfrac{0.5}{3}\,[\,0 + 4(\dfrac{\sqrt{2}}{2}) + 1\,] = \dfrac{1}{6}(1+2\sqrt{2})