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Part 2. Numerical Integration

TABLE 2.1 Important integration formulas

Method Formula Error
Left Sum Approximation ~O(h)
Right Sum Approximation ~O(h)
Midpoint Approximation ~O(h2)
Trapezoidal Approximation ~O(h2)
Simpsonís (1/3) Approximation ~O(h4)
Simple Monte Carlo Approximation ~

 

* ; ** .


Example 1

Suppose we want to evaluate the integral . The definite integral is the area under the curve , above the x-axis and between the lines x = 0 and x = 10. The actual value can be easily obtained as follows:

.

 

Solution. n = 4(h = 2.5)

a)Use it in conjunction with Left sum approximation to integrate the same function for four subintervals:

.

The numerical error and the true percent relative error are:

.

b)Using Right sum approximation:

.

.

c)Using Trapezoidal Rule (four subintervals):

.

.

d)Using Simpson's 1/3 Rule (four subintervals):

.

.

Example 2

Suppose we want to evaluate the following double integral:

.

For the numerical evaluations use both the trapezoidal and Simpson's 1/3 rules. For both cases, use the multiple-application version, with n = 4. Compute percent relative errors for the numerical results.

Solution.

a) Analytically the actual value can be easily obtained as follows:

b)Numerical integration with n = 4:(hx = 1.0; hy = 2.0)

c)Use it in conjunction with Trapezoidal Rule to integrate the same function for four subintervals:

.

d)Using Simpson's 1/3 Rule:

.

Note:

∑ Trapezoidal rule gives the exact value of the integral for linear polynomials;

∑ The Simpson's rule gives the exact value of the integral for polynomials of degree three or less;

∑ An even number subintervals must be utilized to the Simpson's rule;

∑ As the number of subintervals increases, the approximations get better.

 

Problems

1. Integrate the following function both analytically and numerically.

.

For the numerical evaluations use: (a) left sum approximation,
(b)right sum approximation, and (c)midpoint approximation. For all cases use version with n = 4. Compute percent relative errors for the numerical results.

2. Integrate the following function:

.

Note that the true value is I = 0.602297. Use both the trapezoidal and Simpson's 1/3 rules to numerically integrate the function with n = 4.

3. Evaluate the triple integral

(a) analytically, and (b)using Simpson's 1/3 rule with n = 4.
For (b)compute the percent relative error ( ).

4. Integrate the following function both analytically and numerically.

.

For the numerical evaluations use: (a) midpoint approximation,
(b)Monte Carlo integration.

A random sample ( ) of size 20 is given below:

i i i i
0.0105 0.2171 0.2747 0,4150
0.0039 0.5369 0.4442 0.1651
0.3351 0.1957 0.1089 0.8154
0.0332 0.7003 0.6982 0.6855
0.3557 0.9498 0.5643 0.7643

For all cases use version with n = 10. Compute percent relative errors for the numerical results. Discuss your results.



5. The function

can be used to generate the following table of unequally spaced data:

x 0.0 0.12 0.22 0.32 0.36 0.40 0.44 0.54
f(x) 0.20000 1.309729 1.305241 1.743393 2.074903 2.456000 2.842985 3.507297

Evaluate the integral from a = 0 to b = 0.54 using: (a)analytical means, (b)left sum approximation, (c)right sum approximation, and (d)the trapezoidal rule. For (b), (c)and (d),compute the percent relative error ( ).

 


Date: 2016-01-14; view: 775


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