Calculus: Analytic Geometry and Calculus, with Vectors

4.9 Simpson and other approximations to integrals 279

in which n is always an evenpositive integer, we begin by approximating
f(x) over the interval xo < x x2 by the function Q(x) of the form

(4.951) Q(x) = A(x - x1)2 + B(x - x1) + C

whose graph passes through the three points Po(xo,yo), PI(xl,yi), and

P2(x2,y2). Since (4.951) can be put in the form Q(x) = 11x2 + Bix + C1,

its graph is a parabola if A ; 0 and is a line if A = 0. As is easy to guess,

the graph of Q(x) is ordinarily a much better approximation to the arc

P0P2 than the graph consisting of the two straight chords PoP, and P1P2

is, and hence the error term in the Simpson formula is ordinarily much

nearer 0 than the error termin the trapezoidal formula. We find that

x=Q(x) dx =fA (x 3 xi)

+ B(x^2 x1)2+ C(x - xi)

J

`+h

xo L xi-h

so

f

(4.952) x2 Q(x) dx=3 [214h2 + 6C].

The three formulas

yo = Q(xo) = Q(xj - h) = lfh2 - Bh + C

yl = Q(xl) = C

Y2 = Q(x2) = Q(xl + h) = -40 + Bh + C

enable us to determine .4, B, C in terms of yo, yi, y2 It serves our

purpose, however, to add the first and last of the formulas toobtain

yo+y2=2Ah2+2C

and to note that 4y1 = 4C so

Yo + 4yi + Y2 = 2Ah2 +6C.

This and (4.952) give the formula

h

(4.953) 1 xs Q(x) dx = [yo + 4y1 + y2].

Jxo 3

Using (4.953) and analogous formulas, we see that

ff(x)dx e1+3[yo+4y1+Y2],

f

o

f(x)dx=+3

[y+4y+y

xr

f(x) dx =En/2 + 3 [yn-2 +4y.-1 + ynn

Adding these gives the Simpson formula

b

(4.96) f(x) dx = e +

3

[yo + 4y1 +2y2 + 4ys + 2y4 +..

a

+ 4Y.-1 + Y.],