Newton cotes integration method

This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.

1. Newton Cotes Itegression Method Prepared By :- Pabari Shashikant .

2. INTRODUCTION - In numerical analysis, the Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply N ewton–Cotes rules, are a group of formulae for numerical integration(also called quadrature ) based on evaluating the integrand at equally spaced points. -They are named after Isaac Newton and Roger Cotes. -Newton–Cotes formulae can be useful if the value of the integrand at equally spaced points is given.

3. DESCRIPTION It is assumed that the value of a function ƒ defined on [ a ,  b ] is known at equally spaced points x i , for i = 0, …, n , where x  = a and x n  = b . Solved Using Newton-Cotes Formulae There are two types of Newton–Cotes formulae, 1)The "closed" type. 2)The "open" type.

4. the closed newton-cotes formulae In this type which uses the function value at all ʆ points. The closed Newton–Cotes formula of degree n is stated as… b n ∫ f(x) dx = ∑ wi.f (xi) a i =0 where x i = h i + x , with h (called the step size ) equal to ( x n − x ) /  n = ( b − a ) /  n. The w i are called weights .

5. Closed Newton–Cotes Formulae Degree Common name Formula Error term 1 Trapezoid rule 2 Simpson's rule 3 Simpson's 3/8 rule 4 Boole's rule

6. Trapezoid rules Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an nth order polynomial Then the integral of that function is approximated by the integral of that nth order polynomial. Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial,

7. - T he trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area. It follows that …. The function f ( x ) (in blue) is approximated by a linear function (in red).

8. Error analysis The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result. here exists a number ξ between a and b , such that

9. Method Derived From Geometry The area under the curve is a trapezoid. The integral

10. Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: Use single segment Trapezoidal rule to find the distance covered. Find the true error, for part (a). Find the absolute relative true error, for part (a).

11. Solution a)

12. Solution ( cont ) a) b) The exact value of the above integral is

13. Solution ( cont ) b) c) The absolute relative true error , , would be

14. Open Newton–Cotes Formulas Common name step size Formula Error term Degree Rectangle rule, or midpoint rule 2 Trapezoid method 3 Milne's rule 4 No Name 5

15. The Ope n Newton-Cotes formula In this type which not use the function values at the endpoints. b n-1 ∫ f(x) dx = ∑ wi.f (xi) a i =1 The weights are found in a manner similar to the closed formula.

16. rectangular rules The rectangle method also called the midpoint or mid-ordinate rule. It computes an approximation to a definite integral, made by finding the area of a collection of rectangles whose heights are determined by the values of the function. Formula:- (b - a) f1

17. Specifically, the interval over which the function is to be integrated is divided into equal sub intervals of length. The rectangles are then drawn so that either their left or right corners, or the middle of their top line lies on the graph of the function, with bases running along the –axis,,, Giving formula are,,,,, where h=(b - a) / N And Xn=a + nh The formula for above gives for the Top-left corner approximation.

18. Animation Based Rectangular Method Graph Midpoint Approximation Graph

19. Error Analysis For a function which is twice differentiable, the approximation error in each section of the midpoint rule decays as the cube of the width of the rectangle. for some in . Summing this, the approximation error for intervals with width is less than or equal to N=1,2,3 where n + 1 is the number of nodes

20. n terms of the total interval,we know that so we can rewrite the expression : for some in ( a,b ).

21. Application of Simpson’s Rules Space Find the distance of the travel through velocity and interval of time Find the Volume of the solid Calculate amount of earth that must be moved to fill a depression or make a dam.

22. The Velocity v(km/min) of a moped which starts from rest is given at fixed intervals of time t(min) as follows: Estimate Approximately the distance covered in 20 minutes. Sol. If s(km) be the distance covered in t(min), then REAL TIME EXAMPLE OF SIMPSON’S METHOD

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