Page 107 - Schaum's Outline of Theory and Problems of Advanced Calculus

98 INTEGRALS [CHAP. 5

For further examples, see Problems 5.29 and 5.74 through 5.76. For further discussion of improper
integrals, see Chapter 12.

NUMERICAL METHODS FOR EVALUATING DEFINITE INTEGRALS
Numerical methods for evaluating definite integrals are available in case the integrals cannot be

evaluated exactly. The following special numerical methods are based on subdividing the interval ½a; bŠ
into n equal parts of length x ¼ðb aÞ=n. For simplicity we denote f ða þ k xÞ¼ f ðx k Þ by y k , where
k ¼ 0; 1; 2; ... ; n. The symbol means ‘‘approximately equal.’’ In general, the approximation
improves as n increases.

1. Rectangular rule.
ð b
or
f ðxÞ dx xfy 0 þ y 1 þ y 2 þ þ y n 1 g xf y 1 þ y 2 þ y 3 þ þ y n g ð8Þ
a
The geometric interpretation is evident from the figure on Page 90. When left endpoint
function values y 0 ; y 1 ; ... ; y n 1 are used, the rule is called ‘‘the left-hand rule.’’ Similarly, when
right endpoint evaluations are employed, it is called ‘‘the right-hand rule.’’
2. Trapezoidal rule.
x
ð b
f ðxÞ dx f y 0 þ 2y 1 þ 2y 2 þ þ 2y n 1 þ y n g ð9Þ
a 2
This is obtained by taking the mean of the approximations in (8). Geometrically this
replaces the curve y ¼ f ðxÞ by a set of approximating line segments.
3. Simpson’s rule.
ð b
x
f ðxÞ dx f y 0 þ 4y 1 þ 2y 2 þ 4y 3 þ 2y 4 þ 4y 5 þ þ 2y n 2 þ 4y n 1 þ y n g ð10Þ
a 3
The above formula is obtained by approximating the graph of y ¼ gðxÞ by a set of parabolic
2
arcs of the form y ¼ ax þ bx þ c. The correlation of two observations lead to 10. First,
h
ð
2 h 2
½ax þ bx þ cŠ dx ¼ ½2ah þ 6cŠ
h 3
The second observation is related to the fact that the vertical parabolas employed here are
determined by three nonlinear points. In particular, consider ð h; y 0 Þ, ð0; y 1 Þ, ðh; y 2 Þ then
2
2
2
y 0 ¼ að hÞ þ bð hÞþ c, y 1 ¼ c, y 2 ¼ ah þ bh þ c. Consequently, y 0 þ 4y 1 þ y 2 ¼ 2ah þ 6c.
Thus, this combination of ordinate values (corresponding to equally space domain values) yields
the area bound by the parabola, vertical segments, and the x-axis. Now these ordinates may be
interpreted as those of the function, f , whose integral is to be approximated. Then, as illu-
strated in Fig. 5-3:
n
X h x
3 ½y k 1 þ 4y k þ y kþ1 Š¼ 3 ½ y 0 þ 4y 1 þ 2y 2 þ 4y 3 þ 2y 4 þ 4y 5 þ þ 2y n 2 þ 4y n 1 þ y n Š
k¼1
The Simpson rule is likely to give a better approximation than the others for smooth curves.

APPLICATIONS
The use of the integral as a limit of a sum enables us to solve many physical or geometrical problems
such as determination of areas, volumes, arc lengths, moments of intertia, centroids, etc.