trapezoidal and simpson's 1/3 and 3/8 rule

HITARTH M SHAH PRESENTED TO,SEM 4 BATCH 3C PROF. KEYURI SHAH150120119171

GANDHINAGAR INSTITUTE OF TECHNOLOGY

2141905 | CVNM - COMPLEX VARIABLES AND NUMERICAL METHODS

TOPIC: TRAPEZOIDAL RULE AND SIMPSON’S RULE

WHY WE NEED TO USE THIS AND WHEN??!!

• Problem: Find•   dx• We put u=+1 then du=2x dx.• But the question does not contain an x dx term so we cannot solve it using any

of the integration methods we have met so far.• We need to use numerical approaches. (This is usually how software like

Mathcad or graphics calculators perform definite integrals).• We can use one of two methods:1. Trapezoidal rule2. Simpson's rule

TRAPEZOIDAL AND SIMPSON’S FORMULA AND MEANING OF TERMS

Area =dx b=upper limite a=downward limit(bounded by the curves) n= number of total x terms(total divided parts)

h=difference between two adjacent x terms

(if the table is given then find h direct difference and the number of parts(n) given then find h by formula shown)

HOW ITS FORMULA COME??

Recall that we write "Δx" to mean "a small change in x".Now, the area of a trapezoid (trapezium) is given by:

Area= (p+q)So the approximate area under the curve is found by adding the area of the trapezoids. (Our trapezoids are rotated 90° so that their new base is actually the height. So h = Δx.)Area≈ Δx+ Δx+ Δx+ …..

We can simplify this to give us the Trapezoidal Rule,for n\displaystyle{n}n trapezoids:Area≈Δx( + + )Here Δx=h,and we also need =f(a)=f(a+Δx)=f(a+2Δx)……=f(b)

EXAMPLE 1X 7.47 7.48 7.49 7.50 7.51 7.52f(X) 1.93 1.95 1.98 2.01 2.03 2.06

X=7.47 to X=7.52 ,find Area=(?)

Answer: Area=

Here,a=7.47 b=7.52 n=6 h = 7.48-7.47 = 0.01

By using trapezoidal rule,

=0.005[3.99+15.94]

Trapezoidal rulePAGE:8.4

SIMPSON’S RULE(USE WHEN N=EVEN NUMBER)

X 10 11 12 13 14 15 16Y 1.02 0.94 0.89 0.79 0.71 0.62 0.55

Example 2

= =0.3333[1.57+9.4+3.2]=4.7233

=4.7233

a=10 ,b=16 ,h=1 ,n=6 . PAGE:8.12

SIMPSON’S RULE(USE WHEN N=MULTIPLE OF THREE )

• a=0 , b=3 , n=6 , h=? h = Y= F(X) =

EXAMPLE 3:EVALUATE WITH N=6 BY USING SIMPSON’S RULE AND HENCE CALCULATE LOG 2.

X 0 0.5 1 1.5 2 2.5 3

F(X) 1 0.6667 0.5 0.4 0.3333 0.2857 0.25

PAGE:8.21

BY USING SIMPSON’S RULE,

==1.3888

1.3888………..(1)

BY DIRRECT INTIGRATION , log(1+x)

= log 4 =log

= 2log2………(2)

30

FROM EQUATION 1 & 2 ,……2log2=1.3888

=Log 2=0.6944

ALL THE FORMULA’S FOR N=6,

simpson’s rule

simpson’s rule

Trapezoidal rule =

REFRENCE: http://www.intmath.com/integration/5-trapezoidal-rule.php BOOK: CVNM GTU MC GRAW HILL