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

trapezoidal and simpson's 1/3 and 3/8 rule
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1. Hitarth M shah Presented to, sem 4 batch 3c prof. Keyuri shah 150120119171 Gandhinagar Institute of Technology 2141905 | cvnm - Complex Variables and Numerical Methods Topic: trapezoidal rule and simpson’s rule

2. 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: Trapezoidal rule Simpson's rule  

3. 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)  

4. How its formula come?? Recall that we write " Δ x " 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 = Δ x .) Area≈​​​ ​​ ​​ Δx +​ Δx + ​ Δx +​…..   We can simplify this to give us the Trapezoidal Rule, for n\ displaystyle {n}n trapezoids : Area≈ Δ x( ​​​ + + ​​​ ) Here Δ x=h, and we also need ​​=f(a) =f( a+Δx ) =f(a+2Δx) …… =f(b)  

5. Example 1 X 7.47 7.48 7.49 7.50 7.51 7.52 f(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 rule PAGE:8.4

6. simpson’s rule(use when n=even number)   X 10 11 12 13 14 15 16 Y 1.02 0.94 0.89 0.79 0.71 0.62 0.55 X 10 11 12 13 14 15 16 Y 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

7. 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 0.5 1 1.5 2 2.5 3 F(X) 1 0.6667 0.5 0.4 0.3333 0.2857 0.25 X 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

8. BY USING SIMPSON’S RULE, = =1.3888 1.3888………..(1)   BY DIRRECT INTIGRATION , log(1+x) = log 4 =log = 2log2………(2)   3 FROM EQUATION 1 & 2 ,……2log2=1.3888 =Log 2=0.6944

9. 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