The integral \(\mathop \smallint \limits_0^1 (5{x^3} + 4{x^2} + 3x + 2)dx\)

is estimated numerically using three alternative methods namely the rectangular, trapezoidal, and Simpson’s rules with a common step size. In this context, which one of the following statements is TRUE?

Concept:

General Quadrature Formula (G.Q.F):-

\(I = \mathop \smallint \nolimits_a^b f\left( x \right)dx = \mathop \smallint \nolimits_{{x_0}}^{{x_n}} f\left( x \right)dx\)

\( = h\left[ {n{y_0} + \frac{{{n^2}}}{2}{\rm{\Delta }}{y_0} + \left( {\frac{{{n^3}}}{3} - \frac{{{n^2}}}{2}} \right){{\rm{\Delta }}^2}\frac{{{y_0}}}{{2!}} \ldots } \right]\)

Where, h = Step size

n = Number of strips

1) Trapezoidal rule: Taking n = 1 strip at a time and neglecting second and higher-order differences in G.Q.F

\(\mathop \smallint \nolimits_{{x_0}}^{{x_1}} f\left( x \right)dx = h\left[ {1.{y_0} + \left( {\frac{1}{2}} \right){\rm{\Delta }}{y_0} + Neglect} \right]\)

\( = h\left[ {{y_0} + \frac{1}{2}\left( {{y_1} - {y_0}} \right)} \right]\)

\( = \frac{h}{2}\left( {{y_0} + {y_1}} \right)\)

Again,

\(\mathop \smallint \nolimits_{{x_1}}^{{x_2}} f\left( x \right)dx = h\left[ {1.{y_1} + \frac{1}{2}{\rm{\Delta }}{y_1} + Neglect} \right]\)

\( = h\left[ {{y_1} + \left( {\frac{1}{2}} \right)\left( {{y_2} - {y_1}} \right)} \right]\)

\( = \frac{h}{2}\left( {{y_1} + {y_2}} \right)\)

Similarly, \(\mathop \smallint \nolimits_{{x_2}}^{{x_3}} f\left( x \right)dx = \frac{h}{2}\left( {{y_2} + {y_3}} \right)\) 

\(\mathop \smallint \nolimits_{{x_{n - 1}}}^{{x_n}} f\left( x \right)dx = \frac{h}{2}\left( {{y_{n - 1}} + {y_n}} \right)\)

\(I = \mathop \smallint \nolimits_a^b f\left( x \right)dx = \frac{h}{2}\left[ {{y_0} + {y_n} + 2\left( {{y_1} + {y_2} + {y_3} + \ldots {y_{n - 1}}} \right)} \right]\)

\( \Rightarrow I = \frac{h}{2}\left[ {{y_0} + {y_n} + 2\left( {{y_1} + {y_2} + {y_3} \ldots + {y_{n - 1}}} \right)} \right]\)

2) Simpson’s 1/3^rd Rule: If we take n = 2 strip at a time and neglect 3^rd and the higher-order difference in G.Q.F

\(I = \frac{h}{3}\left[ {{y_0} + {y_n} + 4\left( {{y_1} + {y_3} + {y_5} \ldots } \right) + 2\left( {{y_2} + {y_4} \ldots } \right)} \right]\)

3) Simpson’s 3/8^th Rule: If we take n = 3 strips at a time and neglecting 4^th and higher-order difference in G.Q.F.

\(I = \mathop \smallint \nolimits_a^b f\left( x \right)dx = \mathop \smallint \nolimits_{{x_0}}^{{x_3}} f\left( x \right)dx + \mathop \smallint \nolimits_{{x_3}}^{xb} f\left( x \right)dx \ldots \mathop \smallint \nolimits_{{x_{n - 3}}}^{{x_n}} f\left( x \right)dx\)

\(I = \frac{3}{8}h\left[ {{y_0} + {y_n} + 3\left( {{y_1} + {y_2} + {y_4} + {y_5} \ldots } \right) + 2\left( {{y_3} + {y_6} + {y_9}} \right)} \right]\)

4) Rectangle Rule

In the Rectangle rule, we approximate f_|a,b| using a single interpolation point ‘a’. Our polynomial interpolant will thus be a constant polynomial p(t) = f(a), as shown in figure  and we can calculate its area I_R using:

I_R = f(a) ⋅ (b - a)

Thus,

 1) Trapezoidal Rule gives the exact result for a polynomial of degree 1 because we have neglected 2^nd order difference in G.Q.F while the result exceeds from exact value for higher degree polynomials.

2) Simpson’s 1/3^rd Rule gives the exact result for a polynomial of degree 2, while the result exceeds from exact value for higher degree polynomials.

3) Simpson’s 3/8^th Rule gives the exact result for a cubic polynomial.

4) Rectangle Rule gives the exact result for a constant function.