methods for evaluating definite integrals Numerical

methods for evaluating definite integrals Numerical
For definite integrals such as

?_0^1??e^(?-x?^2 ) dx? or ?_0^1??(1-x^3 ) dx
we can’t use the fundamental theorem of calculus to evaluate them since there are no elementary functions that are antiderivatives of e^(?-x?^2 ) or ?(1-x^3 ). The best we can do is to use approximation methods for such integrals. We will see two methods that work reasonably well and yet are fairly simple.
I- Trapezoidal rule
The trapezoidal rule is a numerical method that approximates the value of a definite integral. We consider the definite integral
?_a^b?f(x)dx
We assume that f(x) is continuous on [a,b] and we divide [a,b] into n subintervals of equal length
h=(b-a)/n
using the n +1 points
x_0=a,x_1=a+h , x_2=a+2h,?,x_(n-1)=a+(n-1)h,x_n=b
We can compute the value of f(x) at these points.
y_0=f(a),y_1=f(a+h) , y_2=(a+2h),?,y_(n-1)=f(a+(n-1)h),y_n=f(b)
The trapezoidal rule can be computed as follows:
?_a^b?f(x)dx?( h )/2 [?(_^)y_0 +2y_1+2y_2+?+2y_(n-1)+y_n?_^ ]




Example 1: Estimate by using trapezoidal rule for n=6.
?_a^b?f(x)dx?( h )/2 [?(_^)y_0 +2y_1+2y_2+2y_3+2y_4+2y_5+y_6?_^ ]
h=(b-a)/n=(1-0)/6=( 1 )/( 6 )

x_n y_n Factors Product
0 y_0=1 1 1
0+1?6 y_1=0.9726 2 1.9452
0+2?6 y_2=0.8948 2 1.7896
0+3?6 y_3=0.7788 2 1.5576
0+4?6 y_4=0.6412 2 1.2824
0+5?6 y_5=0.4994 2 0.9988
1 y_6=0.3679 1 0.3679
Sum 8.9415

?_0^1??e^(?-x?^2 ) dx??(1?6)/2×8.9415=8.9415/12=0.745125





Example 2: Estimate by using trapezoidal rule for n=5.

?_a^b?f(x)dx?( h )/2 [?(_^)y_0 +2y_1+2y_2+2y_3+2y_4+y_5?_^ ]
h=(b-a)/n=(2-1)/5=( 1 )/( 5 )=0.2

x_n y_n Factors Product
1 y_0=1.4142 1 1.4142
1.2 y_1=1.6517 2 3.3034
1.4 y_2=1.9349 2 3.8698
1.6 y_3=2.2574 2 4.5148
1.8 y_4=2.6138 2 5.2276
2 y_5=3 1 3
Sum 21.3276

?_1^2??(1+x^3 ) dx?0.2/2×21.3276=2.13276








II- Simpson s rule
If n is an even then Simpson s rule is given by:

?_a^b?f(x)dx?( h )/3 [?(_^)y_0 +4y_1+2y_2+4y_3+?+2y_(n-2)+4y_(n-1)+y_n?_^ ]

Example 3: Estimate by using Simpson s rule for n=4.

?_a^b?f(x)dx?( h )/3 [?(_^)y_0 +4y_1+2y_2+4y_3+y_4?_^ ]
h=(b-a)/n=(1-0)/4=( 1 )/( 4 )

x_n y_n Factors Product
0 y_0=1 1 1
0+1?4 y_1=0.9682 4 3.8728
0+2?4 y_2=0.866 2 1.732
0+3?4 y_3=0.6614 4 2.6456
1 y_6=0 1 0
Sum 9.2504

?_0^1???(1-x^2 ) dx??(1?4)/3×9.2504=9.2504/12=0.7709







Example 4: Estimate by using Simpson s rule for n=6.

?_a^b?f(x)dx?( h )/3 [?(_^)y_0 +4y_1+2y_2+4y_3+2y_4+4y_5+y_6?_^ ]
h=(b-a)/n=(1-0)/6=( 1 )/( 6 )

x_n y_n Factors Product
0 y_0=1 1 1
0+1?6 y_1=1.5042 4 6.0168
0+2?6 y_2=1.7813 2 3.5626
0+3?6 y_3=2.0281 4 8.1124
0+4?6 y_4=2.2626 2 4.5252
0+5?6 y_5=2.4915 4 9.966
1 y_6=2.7183 1 2.7183
Sum 35.9013

?_0^1??e^?x dx??(1?6)/3×35.9013=35.9013/18=1.9945


Estimate the following definite integral by using numerical methods
1.?_1^2?? e?^x/x dx for n=5 2.?_0^1??(1+x^4 ) dx for n=6