TRAPEZOIDAL RULE IN NUMERICAL ANYLISIS.pptx

TRAPEZOIDAL RULE IN NUMERICAL ANYLISIS.pptx

1. TRAPEZOIDAL RULE By  Anjan Kumar Sharma

2. Xo X1 X2 ……………………………….. Xn Q.Explain trapezoidal rule for finding numerical integration. ProoF  Let F(x) be real value continuous function define on close interval [ a,b ]. We want to find the definite integral

3. Area of trapezium(Xo,X1,Y1,Y0)=1/2(Yo+Y1) * h. For this , we divided graph of F(x) into n number of small trapezium with equal height h. Each trapezium has same height h where h=(b-a)/n. = area(trap(X o ,Y o ,Y 1 ,X 1 )) + area(trap(X 1 ,Y 1 ,Y 2 ,X 2 )) + area(trap(X 2 ,Y 2 ,Y 3 ,X 3 )) +………………………………………….. +area(trap(X n-1 ,Y n-1 ,Y n ,X n ))

4. = [1/2(Y o +Y 1 )*h] +[1/2(Y 1 +Y 2 )*h] + [1/2(Y 2 +Y 3 )*h] +………………………..+ +[1/2(Y n-1 +Yn)*h] =h/2[( Y o +Y n )+2(Y 1 +Y 2 +...+Y n-1 )] o

5. Q.Evaluate by using Trapezoidal rule. Soln  F(x) = a= 0,b=6 , n=6 h=(b-a)/n h =(6-0)/6 h=1 We tabulated X with F(x). 6 X 0 1 2 3 4 5 6 F(x) 1 1/2 1/5 1/10 1/17 1/26 1/37

6. =h/2[(Y o +Y 6 )+2(Y 1 +Y 2 +Y 3 +Y 4 +Y 5 )] =1/2[(1+1/37)+2(1/2+1/5+1/10+1/17+1/26)] =1/2[1.02+1+0.4+0.2+0.11+0.07] =1/2(2.8) =1.4 By Using Trapezoidal rule

7. Try it yourself