Numerical integration:
Name of formula:
Trapezoidal Rule :
**Simpson’s 1/3 Rule:
**Simpson’s 3/8 Rule
Weddles’s Rule
Boole’s Rule
Euler Maclaurin Rule
Q. Math.:
Calculate by Trapezoidal Rule approximate value of 3
∫
x4dx by taking seventh equidistant ordinates.-3
Solution :
Upper limit b=3 ; lower limit a=-3 h= b-a/n ; 3-(-3)/6=1
-3 -2 - 1 0 1 2 3 = n =6
3
∫
x4dx = 1(1/2(81+81) + (1+16+0+1+16)) =115 answer-3
x Y=x4
-3 81 y0
-2 16 y1
-1 1 y2
0 0 y3
1 1 y4
2 16 y5
3 81 y6
Simson’s 1/3 law 3
∫
x4dx= h/3[ (y0+yn)+4(y1+y3+y5+----+yn-1)+2 ((y2+y4+y6+----+yn-2)]-3
= ???
**Simpson’s 3/8 Rule 3
∫
x4dx= h*3/8[(y0+yn) + 3(y1+y2+y4+y5+----+yn-1) + 2 ((y3+y6+y9----+yn-3)]-3
Q: Math: class test:
A river is 80 m wide depth d m at distance x m from one bank is given following table. Calculate area of cross section:
X 0 10 20 30 40 50 60 70 80
y 0 4 7 9 12 15 14 8 3
Solution:
h=b-a/n= 80-0/8=10 80
∫
f(x)dx= h3/8[ (y0+yn)+3(y1+y2+y4+y5+----+yn-1)+2 ((y3+y6+y9----+yn-3)]0
# Math: Calculate by any method (NI) an approximate 3
value of
∫
x4dx by taking 7th equidistant ordinates.-3 Solution:
Given that a= -3, b= 3
h=b-3/n= 3-(3)/6=1
x y=x4
-3 81 y0
-2 16 y1
-1 1 y2
0 0 y3
1 1 y4
2 16 y5
3 81 y6
= 1[1/2(81+81) + (16+1+0+1+16)] =115 Simpson’s 1/3 law:
b
∫
ydx= h/3[ (y0+yn)+4(y1+y3+y5+----+yn-1)+2 ((y2+y4+y6+----+yn-2)]a
= 1/3[(81+81) +4(16+0+16) +2(1+1)] =98 Simpson’s 3/8 law:
b
∫
ydx= 3h/8[ (y0+yn)+3(y1+y2+y4+y5+----+yn-1)+2 (y3+y6+y9----+yn-3)]a
= 3/8[(81+81)+3(16+1+1+16)+2(0)
