Ppt about numerical and statistical methods

1. Topic : Numerical Integration & its Application.
Subject: Numerical & Statistical Methods
Name: Hemali Bishnoi (150953131003)
Dhvani Pandya (150953131011)
Shivani Saini (150953131020)
Batch : A3

2. Integration is an important in Physics.
Used to determine the rate of growth in bacteria or to
find the distance given the velocity (s = ∫vdt) as well as
many other uses.
The most familiar practical (probably the 1
st
usage)
use of integration is to calculate the area.
Integration

3. Integration
• Generally we use formulae to determine the integral of a function:
• F(x) can be found if its antiderivative, f(x) is known.
     aFbFdxxf
b
a


4. Integration
• when the antiderivative is unknown we are required to determine
f(x) numerically.
• To determine the definite integral we find the area between the
curve and the x-axis.
• This is the principle of numerical integration.

5. Integration
Figure shows the area under a
curve using the midpoints

6. Integration
• There are various integration methods: Trapezoid, Simpson’s, etc.
• We’ll be looking in detail at the Trapezoid and variants of the
Simpson’s method.

7. Trapezoidal Rule
• is an improvement on the midpoint implementation.
• the midpoints is inaccurate in that there are pieces of the “boxes”
above and below the curve (over and under estimates).

8. Trapezoidal Rule
• Instead the curve is approximated using a sequence of straight lines,
“slanted” to match the curve.
fi
fi+1

9. Trapezoidal Rule
• Clearly the area of one rectangular strip from xi to xi+1 is given by
• Generally is used. h is the width of a strip.
  iiii xxff   11I
)x-(x½h i1i
1...

10. Trapezoidal Rule
• The composite Trapezium rule is obtained by applying the equation
.1 over all the intervals of interest.
• Thus,
,if the interval h is the same for each strip.
 n1-n2102 f2f2f2ffI  h

11. Trapezoidal Rule
• Note that each internal point is counted and therefore has a weight
h, while end points are counted once and have a weight of h/2.
 
)f2f
2f2f(fdxxf
n1-n
2102
x
x
n
0

 h

12. Simpson’s Rule
• The midpoint rule was first improved upon by the trapezium rule.
• A further improvement is the Simpson's rule.
• Instead of approximating the curve by a straight line, we approximate
it by a quadratic or cubic function.

13. Simpson’s Rule
•Diagram showing approximation using Simpson’s
Rule.

14. Simpson’s Rule
• There are two variations of the rule: Simpson’s 1/3 rule
and Simpson’s 3/8 rule.
• The formula for the Simpson’s 1/3,
   n1-n32103
x
x
f4f4f2f4ffdxxf
n
0
 h

15. Simpson’s Rule
• The integration is over pairs of intervals and requires that total
number of intervals be even of the total number of points N be odd.

16. Simpson’s Rule
• The formula for the Simpson’s 3/8,
   n1-n32108
3
x
x
f3f2f3f3ffdxxf
n
0
 h
If the number of strips is divisible by three we can use
the 3/8 rule.

17. Thank You