The Romberg's method is I=I2+31(I2−I1) , where I1 and I2 are obtained by using Trapezoidal rule taking h = 1 and h = 0.5.
Taking h=1, the tabulated values are
x :0 1 2 3 4y=f(x):1 3 2.5 3.6 1.8
Using Trapezoidal rule,
I1=∫04f(x)dx=2h((y0+yn)+2(y1+y2+⋯+yn−1))=21((1+1.8)+2(3+2.5+3.6))=10.5
Taking h=0.5, the tabulated values are
x :0 0.5 1 1.5 2 2.5 3 3.5 4y=f(x):1 4 3 2 2.5 2.9 3.6 4 1.8
Using Trapezoidal rule,
I2=∫04f(x)dx=2h((y0+yn)+2(y1+y2+⋯+yn−1))=20.5((1+1.8)+2(4+3+2+2.5+2.9+3.6+4))=11.7
Now, using Romberg's formula with I1 and I2
I=I2+31(I2−I1)=11.7+31(11.7−10.5)=12.1
Hence approximate value of ∫04f(x)dx is 12.1
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