Question

We can generate the Trapezoidal Rule with help of

(i) General Quadrature Formula (ii) Newton’s Forward Formula

(iii) Stirling Formula (iv) Bessel’s Formula​

Answer 1

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Answer 2

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TO DETERMINE

We can generate the Trapezoidal Rule with help of

We can generate the Trapezoidal Rule with help of(i) General Quadrature Formula

(ii) Newton’s Forward Formula

(ii) Newton’s Forward Formula(iii) Stirling Formula

(iv) Bessel’s Formula

EVALUATION

THE ANSWER IS

We can generate the Trapezoidal Rule with help of

We can generate the Trapezoidal Rule with help of(i) General Quadrature Formula

EXPLANATION

GENERAL QUADRATURE FORMULA is given by :

$\displaystyle \sf{\int\limits_{x_0}^{x_0 + nh} f(x) \, dx } = nh \bigg[ y_0 + \frac{n}{2} \Delta y_0 + \frac{n(2n - 3)}{12} {\Delta }^{2}y_0 + \frac{n {(n - 2)}^{2} }{24} {\Delta }^{2}y_0 + ........ \bigg]$

For TRAPEZOIDAL RULE we put n = 1 & taking the curve through the points

$\sf{(x_0,y_0) \: \: and \: \: (x_1,y_1)}$

as a straight line i.e. a polynomial of first order so that differences of order higher than first become zero.

Thus we get

$\displaystyle \sf{\int\limits_{x_0}^{x_0 + h} f(x) \, dx } = h \bigg[ y_0 + \frac{1}{2} \Delta y_0 \bigg] = \frac{h}{2} \bigg[ y_0 + y_1 \bigg]$

Similarly

$\displaystyle \sf{\int\limits_{x_0 + h}^{x_0 + 2h} f(x) \, dx } = h \bigg[ y_1 + \frac{1}{2} \Delta y_1 \bigg] = \frac{h}{2} \bigg[ y_1 + y_2 \bigg]$

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$\displaystyle \sf{\int\limits_{x_0 + (n - 1)h}^{x_0 + nh} f(x) \, dx } = \frac{h}{2} \bigg[ y_{n - 1} + y_n \bigg]$

Adding these n integrals

$\displaystyle \sf{\int\limits_{x_0}^{x_0 + nh} f(x) \, dx } = \frac{h}{2} \bigg[( y_{0} + y_n ) + 2( y_{1} + y_2 + ..... + y_{n - 1}) \: \bigg]$

Which is known as TRAPEZOIDAL RULE

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