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GATE PI 2017 Official Paper

Option 2 : Parabola

__Explanation:__

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

Simpson’s one-third rule: If we take n = 2 strip at a time and neglect 3rd and the higher-order difference in G.Q.F

- In Simpson's (1/3)rd rule, the curve y = f(x) is approximated by a parabola in every sub-interval.
- The given interval must be divided into an even number of equal sub intervals.
- It corresponds to using second-order polynomials.

\(\mathop \smallint \limits_{{x_0}}^{{x_0} + nh} f\left( x \right)dx = \frac{h}{3}\left[(y_0\;+\;y_n)+4(odd)+2(even)\right]\)

__Important Points__

- Trapezoidal Rule gives the exact result for a polynomial of degree 1 because we have neglected 2nd order difference in G.Q.F while the result exceeds from exact value for higher degree polynomials.
- Simpson’s 1/3rd Rule gives the exact result for a polynomial of degree 2 i.e. parabola, while the result exceeds from exact value for higher degree polynomials.
- Simpson’s 3/8th Rule gives the exact result for a cubic polynomial.
- Rectangle Rule gives the exact result for a constant function.

__Additional Information__

Trapezoidal rule:

\(\mathop \smallint \limits_{{x_0}}^{{x_0} + nh} f\left( x \right)dx = \frac{h}{2}\left[ {\left( {{y_0} + {y_n}} \right) + 2\left( {{y_1} + {y_2} + - - - - {y_{n - 1}}} \right)} \right]\)

Here, the in the interval is divided into n number of intervals of equal width h.

Simpson’s three-eight rule:

\(\mathop \smallint \limits_{{x_0}}^{{x_0} + nh} f\left( x \right)dx = \frac{{3h}}{8}\left[ {\left( {{y_0} + {y_n}} \right) + 3\left( {{y_1} + {y_2} + {y_4} + {y_5} \pm - - - {y_{n - 1}}} \right)} \right] + 2\left[ {{y_3} + {y_6} + \; - - - - {y_{n - 3}}} \right)]\)

Here, the number of subintervals should be taken as a multiple of 3.

Weddle’s rule:

\(\mathop \smallint \limits_{{x_0}}^{{x_0} + nh} f\left( x \right)dx = \frac{{3h}}{{10}}\left[ {{y_0} + {y_1} + {y_2} + 6{y_3} + {y_4} + 5{y_5} + 2{y_6} + 5{y_7} + {y_8} + x} \right]\)

Here, the number of sub-intervals should be taken as a multiple of 6.