82 4. NUMERICAL INTEGRATION AND DIFFERENTIATION

Theorem (Composite Trapezoidal Rule). Suppose f 2 C2[a, b]. Then forsome µ 2 (a, b) we haveZ b

af(x) dx =

h

2

hf(a) + f(b) + 2

n�1Xj=1

f(xj)i� (b� a)h2

12f 00(µ).

Composite Midpoint Rule

If f 2 C2[a, b], then a number ⇠ in (x�1, x1) exists withZ x1

x�1

f(x) dx = 2hf(x0) +h3

3f 00(⇠)

where h =x1 � x�1

2.

Divide [a, b] into n + 2 intervals, n even, h =b� a

n + 2, xj = a + (j + 1)h. Then

apply the Midpoint Rule to successive pairs of intervals.

0 1 0 00 1 1 0

Thus the composite weight pattern is

0� 1� 0� 1� · · ·� 0� 1� 0.

Theorem (Composite Midpoint Rule). Suppose f 2 C2[a, b]. Then forsome µ 2 (a, b) we have

Z b

af(x) dx = 2h

n/2Xj=0

f(x2j)�(b� a)h2

6f 00(µ).