The table below gives values of function F(x) obtained for values of x at intervals of 0.25.

x	0	0.25	0.5	0.75	1.0
F(x)	1	0.9412	0.8	0.64	0.50

The value of the integral of the function between the limits 0 to 1 using Simpson's rule is

Concept:

The Simpson’s rule is given by,

\(\mathop\int\nolimits_{{x_0}}^{{x_0} + nh} f\left( x \right)\;dx = \dfrac{h}{3}[\left( {{y_0} + {y_n}} \right) + 2\left( {{y_2} + {y_4} + \ldots + {y_{n - 2}}} \right) + 4\left( {{y_1} + {y_3} + \ldots + {y_{n - 1}}} \right)\)

h –Width of interval / Step length

y₀, y₁ …y_n – Ordinates corresponding to x₀, x₁, ……, x_n

Calculation:

Given:

h = 0.25

\(\;{\rm{I}} = \dfrac{{0.25}}{3}[\left( {1 + 0.5} \right) + 4\left( {0.9412 + 0.64} \right) + 2\left( {0.8} \right) = 0.7854\)