Question #110537
Compute integral 0 to 4 f(x)dx using the Romberg integral technique on the trapezoidal integrals evaluated by the trapezoidal rule taking h = 1 and h = 0.5. The tabulated
values are given below.
x 0 0.5 1 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 1 4 3 2 2.5 2.9 3.6 4 1.8
1
Expert's answer
2020-04-22T18:42:43-0400

The Romberg's method is I=I2+13(I2I1)I = I_{2}+\dfrac{1}{3}(I_{2}-I_{1}) , where I1I_{1} and I2I_{2} 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    1.8x ~~~~~~~~~~~~~: 0~~~~1~~~~2~~~~~~3~~~~~~~4\\ y=f(x):1~~~3~~~2.5~~~3.6~~~~1.8


Using Trapezoidal rule,

I1=04f(x)dx=h2((y0+yn)+2(y1+y2++yn1))=12((1+1.8)+2(3+2.5+3.6))=10.5I_{1} = \displaystyle\int_{0}^{4}f(x)dx = \dfrac{h}{2}\left((y_{0}+y_{n})+2(y_{1}+y_{2}+\cdots+y_{n-1})\right)\\ =\dfrac{1}{2}\left((1+1.8)+2(3+2.5+3.6)\right)=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     1.8x ~~~~~~~~~~~~~: 0~~~~0.5~~~~1~~~~1.5~~~~~2~~~~~~2.5~~~~~3~~~~~3.5~~~~~4\\ y=f(x):1~~~~~~4~~~~~3~~~~~~2~~~~~2.5~~~~2.9~~~~3.6~~~~~4~~~~~1.8


Using Trapezoidal rule,

I2=04f(x)dx=h2((y0+yn)+2(y1+y2++yn1))=0.52((1+1.8)+2(4+3+2+2.5+2.9+3.6+4))=11.7I_{2} = \displaystyle\int_{0}^{4}f(x)dx = \dfrac{h}{2}\left((y_{0}+y_{n})+2(y_{1}+y_{2}+\cdots+y_{n-1})\right)\\ =\dfrac{0.5}{2}\left((1+1.8)+2(4+3+2+2.5+2.9+3.6+4)\right)=11.7


Now, using Romberg's formula with I1I_{1} and I2I_{2}


I=I2+13(I2I1)=11.7+13(11.710.5)=12.1I = I_{2}+\dfrac{1}{3}(I_{2}-I_{1}) = 11.7 + \dfrac{1}{3}(11.7-10.5) = 12.1


Hence approximate value of 04f(x)dx is 12.1\displaystyle\int_{0}^{4}f(x)dx~ \text{is}~ 12.1


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