The document explains the trapezoidal rule for numerical integration, detailing how to approximate the area under a curve by dividing it into trapeziums. It demonstrates the calculation using a specific function over a defined interval, providing formulas and a sample evaluation with the trapezoidal method. The final result obtained from the calculation is 1.4.

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(Xo,Yo,Y1,X1))
+area(trap(X1,Y1,Y2,X2))
+area(trap(X2,Y2,Y3,X3))
+…………………………………………..
+area(trap(Xn-1,Yn-1,Yn,Xn))

4. = [1/2(Yo+Y1)*h]
+[1/2(Y1+Y2)*h]
+[1/2(Y2+Y3)*h]
+………………………..+
+[1/2(Yn-1+Yn)*h]
=h/2[(Yo+Yn)+2(Y1+Y2+...+Yn-1)]

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).
0
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[(Yo+Y6)+2(Y1+Y2+Y3+Y4+Y5)]
=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