Use Romberg Integration to compute R 3 , 3 {\displaystyle R_{3,3}} for ∫ 0 1 x 2 e − x d x {\displaystyle \int _{0}^{1}x^{2}e^{-x}\,dx}
Solution:
R 1 , 1 = h 1 2 [ f ( 0 ) + f ( 1 ) ] {\displaystyle R_{1,1}={\frac {h_{1}}{2}}[f(0)+f(1)]} R 1 , 1 = 1 2 [ 0 + 1 e ] {\displaystyle R_{1,1}={\frac {1}{2}}[0+{\frac {1}{e}}]} R 1 , 1 = .1839397206 {\displaystyle R_{1,1}=.1839397206}
R 2 , 1 = ( 1 2 ) [ R 1 , 1 + h 1 f ( a + h 2 ) ] {\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]} R 2 , 1 = .1379547904 {\displaystyle R_{2,1}=.1379547904}
R 3 , 1 = ( 1 2 ) [ R 2 , 1 + h 2 ( f ( a + h 3 ) + f ( a + 3 h 3 ) ) ] {\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]} R 3 , 1 = .1475727039 {\displaystyle R_{3,1}=.1475727039}
R 2 , 2 = R 2 , 1 + R 2 , 1 − R 1 , 1 4 − 1 {\displaystyle R_{2,2}=R_{2,1}+{\frac {R_{2,1}-R_{1,1}}{4-1}}} R 2 , 2 = .1226264803 {\displaystyle R_{2,2}=.1226264803}
R 3 , 2 = R 3 , 1 + R 3 , 1 − R 2 , 1 4 − 1 {\displaystyle R_{3,2}=R_{3,1}+{\frac {R_{3,1}-R_{2,1}}{4-1}}} R 3 , 2 = .1507786751 {\displaystyle R_{3,2}=.1507786751}
R 3 , 3 = R 3 , 2 + R 3 , 2 − R 2 , 2 16 − 1 {\displaystyle R_{3,3}=R_{3,2}+{\frac {R_{3,2}-R_{2,2}}{16-1}}} R 3 , 3 = .1526554881 {\displaystyle R_{3,3}=.1526554881}
![{\displaystyle R_{1,1}={\frac {h_{1}}{2}}[f(0)+f(1)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3883df79a3af4959432bcde8510e575edeae24)
![{\displaystyle R_{1,1}={\frac {1}{2}}[0+{\frac {1}{e}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4317821c9fdfbf0f74f6e400a12d90811c6f58)
![{\displaystyle R_{2,1}=\left({\frac {1}{2}}\right)[R_{1,1}+h_{1}f(a+h_{2})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0eb68efb498a0c1bcc0c41162ad941d37272440)
![{\displaystyle R_{3,1}=\left({\frac {1}{2}}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2a9d2c2f22ba9a362f98530065acde7717ad4f)
