**MATH3242.03/COSC3122.03**

**FW03**

**Assignment 2 Due: Mon., Feb.
2, 2000**

1. Use the elementary Trapezoidal Rule, Simpson's Rule and the Midpoint Rule to find an approximation to

(a) ; (b) .

2. Repeat question 1 using the composite form of the three integration rules with n = 4 for the Trapezoidal and Simpson's Rules and n = 2 for the Midpoint Rule (i.e. using the same h value in all cases).

3. Use the composite form of the three rules listed in question 1 to evaluate where f is tabulated below:

x |
1.0 |
1.1 |
1.2 |
1.3 |
1.4 |
1.5 |
1.6 |

f(x) |
2.31 |
2.38 |
2.53 |
2.85 |
3.44 |
4.51 |
6.42 |

4. Find n such that the composite Simpson's Rule will yield a value for to an accuracy of 3 decimal places and calculate this value.

5. Apply Romberg integration to evaluate the integral in question 4.
Calculate 4 values using the composite Trapezoidal Rule, i.e.
calculate R_{k}^{(1)} for k = 1,4 and then apply
Richardson extrapolation to the result.

6. Show that applying Richardson extrapolation to the composite Trapezoidal Rule yields the composite Simpson's Rule. Hint: use equation (4.32) of Burden & Faires (sixth edition).

7. Use the MAPLE worksheet for Algorithm 4.1 (Composite Simpson's Rule) to evaluate the integral (You can download this worksheet from my website if you don't have the cdrom from the text). Take n = 4 as a first guess and then keep doubling n until you are sure you have the result correct to 5 decimal places. Hand in the result you get for each value of n.

**TERM TEST: MON. FEB. 9, 2000**