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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) Simpson’s (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 0.12 0.22 0.32 0.36 0.4 0.44 0.54 f(x) 0.2 1.30973 1.30524 1.74339 2.0749 2.456 2.84299 3.5073

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: 1152

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